Lattice Points (Pitman Monographs & Surveys in Pure & Applied Mathematics.)

Lattice Points

Main Editors

H. Brezis, Universite de Paris R. G. Douglas, State University of New York at Stony Brook A. Jeffrey, University of Newcastle-upon-Tyne (Founding Editor) Editorial Board

R. Aris, University of Minnesota A. Bensoussan, INRIA, France S. Bloch, University of Chicago B. Bollobas, University of Cambridge W. Burger, Universitat Karlsruhe S. Donaldson, University of Oxford J. Douglas Jr, University of Chicago R. J. Elliott, University of Alberta G. Fichera, Universite di Roma R. P. Gilbert, University of Delaware R. Glowinski, Universite de Paris K. P. Hadeler, Universitat Tiibingen K. Kirchgassner, Universitat Stuttgart B. Lawson, State University of New York at Stony Brook W. F. Lucas, Claremont Graduate School R. E. Meyer, University of Wisconsin-Madison J. Nitsche, Universitat Freiburg L. E. Payne, Cornell University G. F. Roach, University of Strathclyde J. H. Seinfeld, California Institute of Technology B. Simon, California Institute of Technology I. N. Stewart, University of Warwick S. J. Taylor, University of Virginia

I

Pitman Monographs and Surveys in Pure and Applied Mathematics 39

Lattice Points P. Erdos, P. M. Gruber & J. Hammer Hungarian Academy of Sciences/Technical University of Vienna/University of Sydney

Longman

NN Scientific& now Technical Copublished in the United States with ,lohn Wiley & Sons, Inc, New York

Longman Scientific & Technical Longman Group UK Limited Longman House, Burnt Mill, Harlow Essex CM20 2JE, England and Associated Companies throughout the world.

Copublished in the United States with John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158

© P. Erdos, P. M. Gruber and J. Hammer 1989 All rights reserved; no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise without either the prior written permission of the Publishers or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, 33-34 Alfred Place, London, WC1E 7DP. First published 1989 ISSN 0269-3666

AMS Subject Classifications. (Main) 11HXX, 52-XX, 05CXX, (Subsidiary) 82A60, 53C65, 65030 British Library Cataloguing in Publication Data

Erdos, Paul, 1913Lattice points. 1. Lattice point geometry I. Title II. Gruber, P. M.

III. Hammer, J.

516.3'5

ISBN 0-582-01478-6 Library of Congress Cataloging-in-Publication Data Erdos, Paul, 1913-

Lattice points/P. Erdos, P. Gruber & J. Hammer. p. cm. - (Pitman monographs and surveys in pure and applied mathematics, ISSN 0269-3666; 39) Bibliography: p. Includes index. ISBN 0-470-21154-7

1. Lattice theory. 2. Geometry of numbers. I. Gruber, Peter M., 1941- . II. Hammer, J. (Joseph) III. Title. IV Series. QA171.5.E73 1989 511.3'3- dcl9

Typeset in 10/12 Times New Roman Printed and Bound in Great Britain at The Bath Press, Avon.

Contents

Preface

List of symbols

1 Equidissectable polytopes 2 Lattice polytopes, lattice point enumerators and a glimpse of algebraic geometry 3 Minkowski's fundamental theorem and some of its relatives 4 Blichfeldt's theorem 5 Successive minima 6 The Minkowski-Hlawka theorem 7 Mahler's selection theorem 8 Packing and covering 9 Packing and covering with balls 10 Crystallography, tiling and Hilbert's 18th problem 11 Geometry of positive quadratic forms: reduction, packing and covering with balls 12 Selected problems of number theory

vi

viii 1

6 14 25 28

34 37 40 55 67 81

13 Visibility 14 Lattice point problems of integral geometry 15 Applications to numerical analysis 16 Lattice graphs 17 Extremal combinatorial problems

96 107 111 116 123 133

References

139

Subject index

176

Author index

180

V

Preface

In this book we have tried to collect geometric, number-theoretic and also combinatorial and analytical results, theories and problems related to lattice points. It is clear that problems of the geometry of numbers comprise a sizeable part of this book, but we have tried to cover more topics dealing with dissection problems, lattice polytopes, packing, covering and tiling problems, mathematical crystallography, visibility, integral geometry, applications to numerical integration, combinatorics, graph theory and several others.

We hope that the book will convince the reader of the many interesting relations of the concept of lattice points to other areas of mathematics and that the great number of implicitly or explicitly stated classical and new problems will induce further research. Since it was our intention that the book should act as an `appetizer' we have included

only a small number of proofs, but we have not hesitated to state heuristic arguments and to give intuitive descriptions. We also make many comments on the results presented, and, in some instances our personal opinion, is made clear. The references are selective, and we have always tried to include more recent ones. If by chance we have omitted some prominent paper in the field of lattice points we ask to be pardoned. In any case readers interested in geometry of numbers may consult the comprehensive volume on Geometry of numbers, the second

edition of which was prepared by G. Lekkerkerker and one of the present authors. Many references are to surveys and monographs on various topics dealing with lattice points which the reader might wish to consult.

We gratefully acknowledge many helpful hints and discussions with colleagues and friends working in the area of lattice points, in particular Professors Coxeter, Danzer, Ewald, Gabor Fejes TOth, Groemer, Hlaw-

ka, Mack, McMullen, Ryskov, Seidel, Shephard, Uhrin and Wills. Professor Ewald helped us in the preparation of section 2.5 and

PREFACE

vii

Professor Shephard made many helpful comments on Penrose tilings. The figures were drawn by Hartwig Sorger. Christian Buchta, Gerhard Ramharter, Dinesh Sarvate and Esther Szekeres assisted us with checking the manuscript. The typing was done by Yit-Sin Choo. P. Erdos P. M. Gruber J. Hammer

To Laszlo Fejes Toth

List of symbols

A 23

A()7 a( 19 a( ) 2 Bd 3

B( ) 8 bd 7

d( ,

L L(

,)10

)j,A,(, )28 J1189

m()89 µ(

C* 30 C( , ) 10 D 23 D( )74,88

d(

L( ),) °( ), L°( ) 6, 125

14

) 18

A( ) 16 b( ) 16 OL( ) , O (), bc() 41 bi( ), oT( ), bc() 42 det 16

) 54

123

1

j

11 37

p81

P()8 PA( ) 21 7!( ) 92 pos 11 R 23 iP 83

P()89 S

11

[d, k, m] 60 (d, M, m) 60

S(

Ed 1

-T 82

E(

7, 123

F() 32

0L(

id 68

V(

11 xd 17

'cO18,21,99

X()6 -e 37

L* 32

15

a()93 BT( ), BcO 41

BL( ), BT( ), e k( ) 42

3, 123

V( ,..., ,) 10 W 23

W;( ) 23

Z2 x()34

x(,)116

1

Equidissectable polytopes

1.1 The investigation of equidissectability of polytopes was greatly stimulated by Hilbert's third problem. First we shall consider a modern

variant of Hilbert's problem in the context of lattices and then a problem due to Hadwiger will be treated. The original problem of Hilbert, which essentially goes back to Gauss, is

specify two tetrahedra of equal bases and equal altitudes which can in no way be split up into congruent tetrahedra, and which cannot be combined with congruent tetrahedra to form two polyhedra which themselves could be split into congruent tetrahedra. (Hilbert [1900])

In order to formulate a general version of Hilbert's problem some precise definitions must be given. Let a class S of subsets of d-dimensional euclidean space E d be given. In particular we shall consider classes of proper polytopes in Ed. These

are finite unions of (compact) convex polytopes with non-empty interiors. A dissection in S of a set S E S is a finite family (Si, . ., S having pairwise disjoint interiors and such that S equals the union of S1, . . ., S,,. In addition to S let a group G of rigid motions in .

Ed be given. Two sets, S, T E S are called equidissectable in S with respect to G or G-equidissectable in S if there are dissections {S1, ..., S}, { T1, . . ., T,) in S of S and T respectively such that Ti = m,(S,) for i c- {1, ., n} for suitable rigid motions m1, ..., Mn E G. The sets S and T are .

.

G-equicomplementable in S if there are G-equidissectable sets U, V E S such that S and U have disjoint interiors and correspondingly for T and V and such that S u U and T u V are G-equidissectable. Papers related to Hilbert's problem typically deal with the following question: Given a class (S of subsets and a group G of rigid motions of 1

LATTICE POINTS

2

Ed, specify necessary and/or sufficient conditions for sets S, T E S to be G-equidissectable or G-equicomplementable.

An old result for the plane due to Farkas Bolyai (the father of the famous Janos Bolyai) and P. Gerwien says the following. Two proper polygons, i.e. proper polytopes in E2, are equidissectable with respect to the group of all rigid motions in the class of all polygons if and only if they have the same area. Unfortunately the corresponding result does not hold for dimensions > 3 as was shown by Dehn, then assistant to Hilbert. In particular, Dehn's result gives a positive answer to Hilbert's third problem. For exhaustive information on results in the context of

Hilbert's third problem we refer the reader to Hadwiger (1957), Boltyanskii (1978), Sah (1979) and the survey of McMullen and Schneider (1983). For results of a related character in the context of the famous

Banach-Tarski paradox, but based on a concept of dissection into disjoint sets, see Wagon (1985).

Let Z' denote the set of all points of Ed with integer coordinates. Z' is called the integer or fundamental lattice in Ed. See figure 1.1. (For the more general concept of a lattice see section 3.1.)

0

0

0

0

0

0

0

0

0

0

0

0

0

0

00

0

0

0

0

0

0

0

0

0

0

0

Figure 1.1 The integer lattice L2

Since we do not distinguish between points and vectors, Z' clearly forms a group. Obviously it can be interpreted as a group of translations: To each u E Z d corresponds the translation

x -x+uforx E Ed. 1.2 Hadwiger (1953, 1957) gave necessary and sufficient conditions for two proper polytopes in Ed to be Zd-equidissectable. In order to formulate his result we need some more definitions.

Let P be a proper polytope and p E P. Then the normalized internal angle a(p, P) of P at p is defined by

EQUIDISSECTABLE POLYTOPES

3

V(P n (pBd + p)) (1.1) V(pBd + p) Here V( ) denotes Lebesgue measure, Bd is the solid euclidean unit ball

a(p, P ) =

li m

P-+0

in Ed and pBd + p = {px + p: x E Bd}. Since the quotient in (1.1) is constant for sufficiently small p > 0, the existence of the limit is obvious. Next define a real functional L on the class of all proper polytopes P in Ed by

L-(P) = E{a(u, P): U E P n Zd}.

(1.2)

L-(P) is the weighted number of points of Z d contained in P and L- is called the weighted lattice point enumerator. For a slightly more general definition and the definition of three more lattice point enumerators see section 2.1. Before giving Hadwiger's criterion for Zd-equidissectability we quote

some of his remarks (1953). Let P be a proper polytope in Ed such that there are translates of P by suitable vectors of Z d which tile Ed, i.e. the translates have pairwise disjoint interiors and their union equals Ed (for more information on tiling see sections 10.4-8). Then

V(P) = L (P).

(1.3)

Thus it is possible to compute the volume of P by `counting' the number of points of Z d contained in P. Assume now that P is a parallelotope all

vertices of which belong to Zd. Then (1.3) holds. We may express L(P) in the form L (P) = E(Ecr(u, P)),

(1.4)

where the inner sum is extended over the points u of Z d contained in

the relative interiors of the i-dimensional faces of P. Denote the number of these points by Ni. It follows from (1.4) that L (P) is simply the sum extended over the values of 2'-dN1. This gives the formula of Hofreiter (1933a) for the volume of P: d

V(P) = E2i-dN. ;=o

Hadwiger's criterion for Z d-equidissectability says that two proper polytopes P, Q in E d are Z d-equidissectable in the class of all proper

polytopes in Ed if and only if L-(P + x) = L-(Q + x) for all x E Ed with 0 < x; < 1 for i e { 1, ., d}. (The coordinates of x E E d are denoted x 1, ... , Xd. Although x is always considered as the column with entries x1, ..., Xd, we shall write (X1, ..., Xd) for x in some .

.

LATTICE POINTS

4

instances.) For an example of a parallelogram equidissectable to a square, see figure 1.2. 0

0

0

which

is

ZZ-

0

0

Fi$ure 1.2 Z2-equidissectability (After Hadwiger (1957), p. 73)

Hadwiger (1953) showed that any two proper polytopes which are

Zd-equicomplementable are also Zd-equidecomposable in the class of all

proper polytopes in E'. Thus for the class of proper polytopes in E d the concept of Z d-equidissectability and Z d-equicomplementability coincide. Coincidence Of these two concepts holds for many classes of polytopes and many groups (see the references cited in section 1.1). For a case where this does not hold see the following subsection.

1.3

Let U denote the group of transformations of the form

x --*Ax+ a

for xEEd,

where A is a d x d matrix with integer elements and determinant ±1 and u E Zd. Each such transformation maps Zd onto itself. By a proper convex lattice polytope in E d we understand a convex polytope with nonempty interior all vertices of which are contained in Zd.

Betke and Kneser (1985) proved that two proper convex lattice polytopes are U-equicomplementable in the class of all convex lattice polytopes if and only if they have equal volume. Unfortunately the concepts of U-equidissectability and Uequicomplementability do not coincide for the class of proper convex

EQUIDISSECTABLE POLYTOPES

5

lattice polytopes. The following is an example of Betke (1985): The simplices in E3 with vertices o = (0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 3) and

o,

(1, 0, 0),

(0, 1, 0),

(0, 0, 3)

respectively

are

U-equi-

complementable but not U-equidissectable in the class of all proper convex lattice polytopes in E3.

2

Lattice polytopes, lattice point enumerators and a glimpse of algebraic geometry 2.1 In this section we consider relations between the number of points of the integer lattice contained in a lattice polytope and its volume and surface area. We also exhibit the connection of lattice polytopes and toric varieties.

A finite family of simplices in Ed is a (finite) simplicial complex if each face of a simplex of the family also belongs to the family and if the intersection of any two simplices of the family is a face of both of them. The 0-dimensional simplices are called vertices. A polytope in E' (or a polygon in case d = 2) is the union of a simplicial complex. The latter forms a simplicial decomposition of the polytope. Equivalently one may define a polytope as a finite union of convex polytopes of dimensions d. A polytope is proper if it is the closure of its interior or if it can be

represented as a finite union of convex polytopes of dimension d, cf. section 1.1. Call a polytope P a lattice polytope if for a suitable simplicial decomposition of P the set of vertices consists precisely of the points of Z d contained in P.

The Euler characteristic of a simplicial complex is the number of its 0-dimensional simplices (= vertices)

minus the number of

its 1-

dimensional simplices (= edges) plus the number of its 2-dimensional simplices, minus etc. For a polytope P the Euler characteristic x(P) can be uniquely defined as the Euler characteristic of a simplicial decomposition of P. For example the three polygons in figure 2.1 have Euler characteristics 1, 2 and 3 respectively.

For a polytope P let L(P), L'(P) and L°(P) denote the number of points of Z d contained in P, on the boundary of P and in the interior of

P, respectively. Further let L_(P) be defined by equations (1.1) and (1.2).

This defines functionals L, L*, L°, L_ on the class of all

polytopes in Ed, so-called lattice point enumerators. 6

LATTICE POLYTOPES AND LATTICE POINT ENUMERATORS

7

2.2 The values of the lattice point enumerators of a lattice polytope and its volume and surface area are closely related.

We consider the case d = 2 first. As may be expected, the results in this case contain dimensional results.

much more information than related higher-

Let P be a lattice polygon of area A(P). Consider a simplicial decomposition of P whose set of vertices is Z2 n P. Let E(P) denote the number of edges of the simplicial decomposition on the boundary bd P of P, where an edge which is contained in the closure of the interior of P is counted once and all other edges twice. A general result of Hadwiger and Wills (1976) says that

L(P) = A(P) + 'E(P) + X(P).

(2.1)

The reader may verify this for the lattice polygons in figure 2.1.

0

(a) polygon but not a lattice polygon

0

0

0

0

(b) lattice polygon

0

(c) proper lattice polygon

Figure 2.1 Lattice and non-lattice polygons

Several particular cases of (2.1) deserve mention. If P is a proper lattice polygon then (2.1) implies the case d = 2 of a formula of Reeve (1957):

L(P) = A(P) + 1L'(P) + x(P) - ix(bdP).

(2.2)

(Note that in Reeve's notation the Euler characteristic is -X(P).) If the boundary of a proper lattice polygon P is the disjoint union of closed Jordan polygons, then X(bd P) = 0 and (2) implies

L(P) = A(P) + zL'(P) + x(P).

(2.3)

A particular case of (2.3) arises when the boundary of a proper lattice polygon P consists of a single closed Jordan polygon. Then we obtain the well-known formula of Pick (1899)

LATTICE POINTS

8

L(P) = A(P) + ZL'(P) + 1.

(2.4)

Among the numerous proofs of Pick's theorem we just mention Coxeter

(1969) and Liu (1979). An application of Pick's theorem to lattice simplices is due to Reznick (1986).

Since each line segment connecting two points of Z2 has length at least 1, proposition (2.1) yields for a lattice polygon P the inequality

L(P) , A(P) + 1B(P) + X(P).

(2.5)

Here the `perimeter' B(P) of P is determined such that each edge on the boundary of P which is contained in the closure of the interior of P is counted twice. To obtain a corollary of (2.5) consider a plane compact convex set C. Let L(C) denote the number of points of Z2 contained in C. If L(C) = 0, inequality (2.5) trivially holds. Otherwise consider the convex hull of the points of Z2 in C. Then (2.5) implies that

L(C) , A(C) + ' P(C) + 1, a result first proved by Nosarzewska (1948). Here P(C) is the perimeter of C. Let P be a lattice polygon and let t E E 2, t f Z2 be chosen. Hadwiger and Wills (1976) give the following somewhat surprising upper bound for the number of lattice points in the translate P + t of P:

L(P + t) , L(P) - X(P). For any value the Euler characteristic can assume there are lattice polygons for which equality holds.

Let us consider the case d , 2. The example of the tetrahedra S with vertices o, (1, 0, 0), (0, 1, 0), (1, 1, m), where m is a positive integer, shows that it is not possible to express V(S) as a linear combination of L(Sm), L*(S,,,) and X(S,,,) (= 1) with real coefficients.

Let P be a proper lattice polytope in Ed. In order to determine V(P) Reeve (1957, 1959) had the idea of considering besides P the lattice polytopes 2P (= {2x: x E P}), 3P, .... The results of Reeve concern the cases d = 2, 3 only. Macdonald (1963) extended and proved them for all d: For d = 2 we have the formulae

A(P) = (L(P) - zL (P)) - (X(P) -'X(bdP)) 2A(P) = L(2P) - 2L(P) + X(P),

A(P) = L'(P), and for d = 3 we obtain

6V(P) _ (L(2P) - ' L'(2P)) - 2(L(P) - ' L'(P)) 2

+ (x(P) - 'X(bd P)),

2

(cf. (2.2)),

LATTICE POLYTOPES AND LATTICE POINT ENUMERATORS

9

6V(P) = L(3P) - 3L(2P) + 3L(P) - X(P), 6V(P) = L(2P) - 2L'(P). For general d the formulae are (d - 1)d! 2

V(P) = (L((d - 1)P) - 'L ((d - 1)P))

(d +

1

1)(L((d - 2)P) - 1L'((d - 2)P)) + . (-1)d-2(

d

d - 12 )((L(P)

- L '(P))

+ (-1)d-'(X(P) - 'X(bd P)),

d!V(P) = L(dP) - d)L((d - 1)P + L(P) + (-1)dX(P),

+

21)d!V(P) (d

= L-((d - 1)P) - (dl + (-1)d-2(d

1 ) L-((d

- 2)P) + .. .

- 2)L (P).

An inequality of Betke and McMullen (1985) relates V(P) and L(nP) for a convex lattice polytope P with non-empty interior:

(n + d

+

n+

d-1 2

(d!V(P) - 1)

(d odd)

d

(n+d)+1{(n+)(n_1+f)}(d!V(P)_1) d

L(nP)n+d-1)d!V(P)+(n+d

d

(d even)

d

d-1

1).

For a large body of further geometric properties of lattice polygons and lattice polytopes, we refer to the thesis of Rabinowitz (1986).

2.3 The functionals L, L', L°, L^ and related other functionals have many interesting properties, for which the reader is referred to McMullen (1975), Ehrhart (1977), Wills (1978, 1980, 1982), Betke and Wills (1979), McMullen and Schneider (1983) and Gruber and Lekkerkerker

LATTICE POINTS

10

(1987). Here we shall state only one particularly appealing result. In

order to appreciate this result we first state a classical theorem of Minkowski (1903); see also Bonnesen and Fenchel (1934), p. 40, or Leichtweiss (1979), p.162. For subsets X, Y of E I and A real define

X+Y={x+y:xEX,yEY}, AX

= {Ax: X E X}.

Let us give an example: assume that C is a convex body in Ed, that is a compact convex subset of Ed with non-empty interior. Then for A > 0 the set C + ABd is called the parallel body of C at distance A. It is easy to see that C + ABd is the union of all balls of radius A and centres in C.

Minkowski's theorem is the following: Let C1, ..., C" be n compact convex sets in Ed. Then there are coefficients V(Ci ..., C,d), i1i ... id E {1, ..., n}, which are symmetric in the indices i1, ..., id, such that for Al, ..., An , 0 the following equality holds:

+.."C') =

+ V(A1C1

n

E

. . .

V(C,,,

.

.

I

it. .id=1

Cid))'iI

.

.

. Aid.

The coefficient V(Ci,, ... , Cid) is called the mixed volume of C,, , ... , Cid .

By analogy with this a result of McMullen (1977) which in essence was proved also by Bernstein (1976) shows that for convex lattice polytopes P1, ., P. in Ed there are coefficients L(P1, i1, ..., P", in), il, ..., in E {0, ..., d} such that for k1i ..., kn c {0, 1, 2, ...} we .

.

have

L(k1p1 + ... + knPn) _

L(P1, il.... , Pn,

in)kl'...kn

.

Similar results hold with L replaced by L', L°, L_.

2.4 For P a lattice polytope Betke and McMullen (1985) consider the power series

L(P, t) = 1 + 7L(nP)t".

,1

If P is a proper lattice polytope, then L(P t) = C(P, t) (1 - t)d+i

LATTICE POLYTOPES AND LATTICE POINT ENUMERATORS

11

where C(P, t) is a suitable polynomial in t of degree at most d + 1 with integer coefficients; see Ehrhart (1967). Betke and McMullen give more precise information on C(P, t) which yields refinements and generalizations of earlier results of Macdonald (1971), Wills (1978) and Stanley (1980).

2.5 Since about 1970 it has become clear that there exists a close relationship between the theory of lattice polytopes and algebraic geometry, centred around the concept of a `toric variety'. In the following we will describe this relationship without giving too many details or exact definitions.

Let 01), . . ., u(k) be primitive points of Zd, i.e. o and u(') are the only points of Zd on the line segment with end points o, u(') for i E {1, ,

k}. Consider the cone

S = pos{u(1),

. .

., u(k)}

(= (Alu(1) + . . + Xku(k): Al, . . ., Ak -- 0})

and the `dual' cone S V of S,

Sv _ {x: (x, y) % 0 for all y E S}. (Here (,) denotes the inner product in Ed.) The set of all Laurent polynomials in d variables z 1, ... , zd , an,...naZ1

nd

a,

where the coefficients an,- nd are in an algebraically closed field K and the summation is over finitely many (n1, . . ., nd) E Sv n Zd, forms a ring Rs". The maximal (prime) ideals of Rs" represent points (irreduci-

ble curves, surfaces, etc.) of an `affine variety' Xs". If S is a cell complex of cones S, i.e. a fan, the affine varieties Xs,, Xs;' of any two cones S1, S2 E A are `glued together' by using the `natural injections' Rs," - R(s,r,s,)". The result is called a toric variety, X,r. If, for example, S1 = pos({(1, 0), (0, 1))),

S2 = pos({(0, 1), (-1, -1)1),

S3 = pos({(1, 0), (-1, -1)}), the affine varieties Xs; , Xsz , Xs, , are affine planes which are `charts' of the projective plane P2 = X,, (where d = {S1, S2, S3, pos{(1, 0)}, pos{(0, 1)}, pos{(-1, -1)}, o}). Toric varieties were introduced by Demazure (1970), Kempf et al.

LATTICE POINTS

12

(1973) and Oda (1978). (See also the survey of Danilov (1978).) There are many properties of the complex S and the toric variety X,, related to each other. Here are some examples:

b X, is compact (if K = C); 1. U{S: S E S} = Ed 2. S is strongly polytopal, that is, there is a lattice polytope P such that

S _ {S = pos{u('), . . , u(k)}: {u(), . ., u(k)} are the ver-

C=> Xy is projective;

.

tices of a face of P}

3. S is regular, that is each d-dimensional a X y has no singularities. cone of S is represented in the form pos{u(1),

...,

u(d)} where det{u(i>,

...,

u(d)} = ±1.

Toric varieties have been used by Stanley (1980) in the solution of

McMullen's conjecture on f-vectors of polytopes and by Teissier (1980/81) for a new proof of the Fenchel-Alexandrov inequalities on mixed volumes. On the other hand results on lattice polytopes were used for problems of algebraic geometry dealing with toric varieties, see for example Oda (1978), Danilov (1978), Batyrev (1982), Voskresenskii and Klyachko (1985), Ewald (1986) and the comprehensive monograph of Oda (1988). Apparently these connections are of increasing importance in contemporary convexity theory.

2.6

We close this section with some remarks on embedding of regular polytopes into V.

Apparently Lucas (1878) was the first to note that an equilateral triangle cannot be embedded into Z2 (here embedding means that all its vertices are in Z2). Scherrer (1946) showed that the square is the only regular polygon that can be embedded into Z2. More generally it was proved by Schoenberg (1937) that the regular triangle, the square and the regular pentagon are the only regular polygons that can be embed-

ded in a suitable Zd. He also showed that a regular d-simplex can be embedded in a suitable Z d' if

1. d is even and d + 1 is a perfect square;

2. d=3mod4;or

3. d is of the form (2h +1)2 + (2k + 1)2 - 1, h, k E {0, 1, 2, ...}. Finally Patruno (1983) completely determined which regular polytopes can be embedded in a suitable Zd. It remains an open question which

LATTICE POLYTOPES AND LATTICE POINT ENUMERATORS

13

semi-regular polytopes can be embedded in a suitable Z1. (For definition of semi-regular figures see Fejes Toth (1964).) For further embedding theorems, see Rabinowitz (1986).

3

Minkowski's fundamental theorem and some of its relatives

3.1 The geometry of numbers has a long history. The first sporadic results appeared in the work of Kepler (1611) and Lagrange (1773). Later contributors who emphasized the geometric aspect are Gauss (1831), Dirichlet (1850) and Klein (1895/96), whereas the numbertheoretic and arithmetic aspect is predominant in the papers of Hermite (1850) and Korkin and Zolotarev (1872, 1873, 1877). Systematic study of the subject started in the decades after 1880 with Fedorov (1885), the basic work of Minkowski (1896, 1907, 1911) and the investigations of Voronoi (1908(a), 1908(b), 1909, 1952). Many results and concepts of the modern geometry of numbers have their origin in the contributions

of Minkowski. It was Minkowski who coined the name geometry of numbers for this new branch of mathematics linking number theory and geometry. Important later contributions are due to Siegel, Furtwangler (mainly unpublished), Davenport, Venkov, Delone, Hajos, their students and a number of other living mathematicians. In this and in some of the following sections we will present basic ideas, results and open

problems of the geometry of numbers. In general, we will present geometric versions of the results chosen. This section deals with a simple but fundamental theorem of Minkowski, which is generally considered

as the central result of the geometry of numbers. Its importance lies in the wide range of interesting applications.

A lattice L in Ed is the set of all linear combinations with integer coefficients of d linearly independent vectors (figure 3.1 shows an example for d = 2). These vectors form a basis of L and the absolute values of their determinant is called the determinant d(L) of L. The parallelotope generated by the basis vectors is called a fundamental parallelotope of L. Its volume is equal to d(L). Note that there are countably many bases of L. In this book we shall not distinguish between vectors and points. Thus we may call the vectors of L (lattice) 14

MINKOWSKI'S FUNDAMENTAL THEOREM

15

points. An example of a lattice is the integer lattice Z' introduced in section 1.1.

r

°

° °

° ° °

Figure 3.1 Lattice in E2

By a convex body in E d we mean a compact convex subset of E d with non-empty interior. V( ) and S( denote Lebesgue measure and ordinary surface area measure in Ed. In case d = 2, we write A( ) (for area) and P( ) (for perimeter) instead.

3.2 The fundamental theorem (of Minkowski (1893)) or Minkowski's convex body theorem says the following. Let C be a convex body, symmetric about the origin o, and let L be a lattice in Ed. If V(C) ? 2dd(L), then C contains at least one pair of points ±p * o of L (see figure 3.2). If C is not a parallelohedron (see section 10.7 for a definition), there is a constant x < 2d depending on C such that the conclusion of the theorem still holds if V(C) % ad(L). For parallelo-

hedra, no such refinement exists. For some historical remarks see section 3.3 below.

0 °

0

0

0

0

0

0

0

0

C

O

O

0

0

°

0

0

Figure 3.2 The fundamental theorem. V(C)

4d(L)

LATTICE POINTS

16

In contrast with the precise information on the equality case in the fundamental theorem, in many inequalities, asymptotic bounds or, in

extremal problems of the geometry of numbers, the best possible constants or the extremal configurations, sets or lattices are not known.

We will sketch a proof of the fundamental theorem which clearly exhibits its geometric background. Let C be an o-symmetric convex body and L a lattice in Ed such that o is the only point of L in C. Then the convex bodies

ZC+p:peL are pairwise disjoint. The system of these convex bodies forms a so-called lattice packing of (2)C. The density of this packing is the proportion of space covered by the bodies or, more precisely, the quotient V((z)C)

d(L) (Packing is treated in more detail in chapter 8.) Since the bodies of the packing are disjoint the density of the packing is less than 1. This yields V(C) < 2dd(L), concluding the proof. For later purposes we shall give a different version of Minkowski's

convex body theorem. Let C be a convex body containing o in its interior or a star body, i.e. a closed subset of Ed with the origin as an interior point and such that each ray starting at o meets its boundary in at most one point. A lattice is admissible for C if it contains no interior point of C except o. Define the critical determinant or lattice constant O(C) of C by A(C) = inf {d(L): L is an admissible lattice for C}. Now the fundamental theorem can be stated in the following form. Let C be an o-symmetric convex body. Then V(C) < 2dO(C).

(3.1)

3.3

Among the numerous applications of Minkowski's fundamental theorem we shall present two classical ones due to Minkowski himself.

For other applications we refer to Cassels (1972) and Gruber and Lekkerkerker (1987).

Let q(x) = Iaikxixk be a real positive definite quadratic form in d variables and with discriminant b(q) = det(aik). For a > 0 the set {x: Za1kxIxk < CV)

MINKOWSKI'S FUNDAMENTAL THEOREM

17

is an ellipsoid with centre o and volume ad12Kd/S(q)1/2, where Kd is the volume of the solid euclidean unit ball in E d. Thus for a = (46(q)/xd)1/d

the ellipsoid contains a pair of points ±u * o of Zd by the fundamental theorem. This shows that the minimum of q(u) for integer values of the variables not all 0 is at most (46(q)) 1/d xdz

This result of Minkowski (1891) considerably improved a theorem of Hermite (1850). For Minkowski's original proof the fundamental

theorem was not yet available, but the idea behind his geometric proof led him to the discovery of the lattice point theorem soon afterwards. The theorem was formulated two years later in a letter of Minkowski addressed to Hermite (Minkowski (1893)). Minkowski's bound for the minimum of positive definite quadratic forms was essentially lowered by the work of Blichfeldt (1929) and in recent years by Kabatjanskii and Levenstein (1978). For a more detailed discussion of this problem in the

equivalent form of the problem of upper bounds for the density of lattice packings of balls we refer the reader to chapter 9. A second application of the fundamental theorem will give the linear form theorem of Minkowski (1896), §37: Let 11, ..., ld be d real linear

forms in d real variables. We require that the absolute value of the

..., ld, say 6, is positive. For reals

determinant of the coefficients of l1,

r1, ..., . rd > 0 the set

{x: 111(x)I < r1i ..., Ild(x)l < id}

is an o-symmetric parallelotope of volume 2 d tl .

.

id/S. If Z1

rd -- 6,

it contains at least one pair of points ±u =f o of Zd by Minkowski's fundamental theorem. Hence the system of inequalities Ill(u)I < 'G1,

.

.

., Ild(u)I < td

has at least one solution u = (u1, .,ud) with integers u1, . . .,ud not all 0 provided the real numbers i1, ..., id > 0 satisfy it Td 6. The linear form theorem has attracted a good deal of attention. The famous theorem of Hajds (1942) which will be discussed in geometric disguise in chapter 8 describes the systems of linear forms 11, ..., ld .

such that for suitable reals 7-1, of inequalities

.

..., rd > 0 with TI .

. Td = 6 the system

Il1(u)I < r,, ..., Ild(u)I < rd

has no solution u = (ul,

ui=.

=ud=0.

.

.

.,ud) with integers u1, ...,ud except for

Mordell (1936) considered a sort of converse problem for the linear

LATTICE POINTS

18

form theorem. Denote by K(d) the supremum of all numbers K > 0 with the following property: let 11, ... , Id be any system of d real linear forms in d variables and with absolute value of the determinant, say 6, positive. Then there are numbers Ti, ..., Td > 0 with T1 Td = KS such that the system of inequalities I11(u)I < T1,

.

.

., Ild(u)I < 'rd

has no solution u = (u I, . . ., U d) with integers u 1i ... , U d except for = ud = 0. It was shown by Szekeres (1936(a)) that U) = K(2) = i + 2U5 = 0.732606....

Alternative proofs and refinements are due to Sziisz (1956), Suranyi (1971) and Gruber (1971). Szekeres (1936(b)) and Ko (1936) proved that K(3) > 4. A local result of Ramharter (1980) supports the conjecture of Gruber that K(3) = ; cos

2, cos 7

7

= 0.578416.

...

Gruber and Ramharter (1982) proved that K(4) > 1116. Hlawka (1950(a)) and Gruber and Ramharter gave the following bounds for K(d): 2-0.5d2+o(d2) =

d (d!)22d(d+1)/2

< K(d)
+oo. For further results see Gruber (1971), Ramharter (1980, 1981) and Gruber and Ramharter (1982).

In sections 10.8 and 12.3 we will return to this subject. An inhomogeneous counterpart of the converse problem was considered by Sawyer (1966) and Bambah et al. (1986).

3.4 In the following subsections we shall present generalizations and refinements of the fundamental theorem. This subsection contains results of a general character whereas in subsequent subsections the case d = 2 and the case of the integer lattice will be treated in more detail. Van der Corput (1935) proved the following result, announced earlier

by Blichfeldt (1921): let C be an o-symmetric convex body and L a lattice in Ed such that V(C) , k2dd(L) for a positive integer k. Then C contains at least k pairs of points ±p(1), ..., ±p(k) * o of L. For a slightly more precise result of this type we refer to White (1963) and Dumir and Hans-Gill (1977). If in van der Corput's theorem d(L) is replaced by d(C, L), where d(C, L) (-- d(L)) is the volume of the subset of a fundamental parallelotope of L covered by the sets 2 C + p:

MINKOWSKI'S FUNDAMENTAL THEOREM

19

p c- L, the conclusion still holds. This result follows from a theorem of Uhrin (1980).

Let C be an o-symmetric convex body and L an admissible lattice for C. Then C is contained in an o-symmetric convex polytope having at most 2d+1 - 2 facets (i.e. faces of dimension d - 1) and for which L is

still admissible. Hence one may replace the constant 2' in the fundamental theorem by 2da, where a (-- 1) is the supremum of the quotient V(C)/V(P) extended over all o-symmetric convex polytopes P containing C and having at most 2d+1 - 2 facets. Results of this type are due to van der Corput and Davenport (1946).

As we have pointed out in section 3.2 the fundamental theorem is related to a certain lattice packing. Since lattice packings are `periodic sets' it seems natural to use multidimensional Fourier series for their investigation. A classical refinement of Minkowski's convex body theorem due to Siegel (1935) is based on Parseval's theorem for Fourier series. Related results were given by Cassels (1947), Hlawka (1944,

1982) and - in some sense - by Uhrin (1981). Siegel's theorem is of interest for algebraic number theory - see Eichler (1963). If C is a convex body with o in its interior, the asymmetry coefficient a(C) of C is defined by a(C) = min {a , 1: -C C aC}.

Many other `measures of asymmetry' have been investigated in recent years. Earlier results in this area were carefully reviewed by Grunbaum (1963).

Let C be a convex body containing o as an interior point and let L be a lattice in Ed. If

V(C) > ((1 + a)d(1 - (1 -

CV-1)d )d(L)

where a = a(C), then C contains at least one point p * o of L according to Sawyer (1954).

Minkowski's fundamental theorem gives, in terms of volume and symmetry, a sufficient condition for a convex body to contain a point p * o of a lattice. The extensions of the fundamental theorem discussed so far are of a similar type. From the point of view of applications in integer programming there is interest in criteria which guarantee that a

convex body contains points of a lattice, and which make use of geometrical quantities different from volume and symmetry. Also from a more geometric and aesthetic viewpoint, working towards results of this

type is desirable. Unfortunately the known results of this type either concern the case d = 2 or apply in many cases to the integer lattice or some other lattices of very particular types. In the next two subsections we consider a series of theorems of this sort. The last subsection of this

LATTICE POINTS

20

chapter deals with results not directly related to Minkowski's theorem.

We conclude this subsection with a conjecture of Ehrhart (1955(b), 1964): a convex body C with centroid o contains at least one pair of points ±p * o of a lattice L, provided V(C) > (d d! l)d d(L).

Here the coefficient of d(L) may be replaced by a smaller one depending on C except in the case when C is a simplex. The conjecture

was proved by Ehrhart for d = 2 and remains open for d > 3. A common generalization of Ehrhart's theorem and the fundamental theorem for d = 2 is due to Scott (1976). In addition to Ehrhart's conjecture we ask whether the condition

V(C) ?

k(d + 1)d

d!

d(L).

for a positive integer k implies that C contains at least k distinct pairs of

points ±p(1), ..., ±p(k) of L\{o}.

3.5

Let d = 2. We first state criteria employing area and symmetry. Then results making use of the affine perimeter and the curvature are mentioned.

For the plane, Sawyer (1955a) gives the following refinement of his result cited in section 3.4. Let C be a convex body which contains o in

its interior and has symmetry coefficient a = a(C) and let L be a lattice. Then each of the following conditions implies that C contains a point * o of L:

1. A(C) ? (3a2 - 2a + 3 - 2(a - 1)(2ca2 + 2)'/2)d(L) and

1 (2a - (2a2 - 4a - 2)'/2)d(L) and 3 zd(L).

This clearly implies Minkowski's fundamental theorem. If o is the centroid of a convex body C, then C has at least 3 chords of symmetry, see e.g. Grunbaum (1963), p.254. Hence the theorem of Ehrhart cited in section 3.4 above is a consequence of Arkinstall's result.

Assume now that C is an o-symmetric convex body such that the boundary of C has continuous curvature K. Let the boundary be parametrized by the arc length s. Then the affine perimeter PA(C) of C is defined by

PA(C) =

J

where the integral is extended along the boundary. The affine perimeter and the corresponding concept of affine surface area of a convex body

in dimensions >3 are invariant with respect to equi-affinities. An equi-affinity is a volume-preserving affinity. The concepts of affine perimeter and affine surface area are due to Blaschke and Pick. For more information we refer the reader to L. Fejes Toth (1972). Making use of results of van der Corput and Davenport (1946), Fejes T6th (1972) proved the following. Let C be as in the last paragraph and let L be a lattice. If PA(C) , 288d(L), then C contains a pair of points p * o of L. The constant 288 is best possible. See also Groemer (1959). The results of this section presented so far have made use of affine or

equi-affine invariants. From the point of view of the geometry of numbers, results involving affine or equi-affine invariants are preferable,

but there are also several interesting results making use of isometric invariants. As an example of a result based on an isometric invariant we give a theorem of Groemer (1961(b)). Let C be an o-symmetric convex

body such that the curvature of the boundary of C exists and has positive infimum and supremum, say 1/R, 1/r. Then C contains a pair of points p 0 o of any lattice L of determinant 1 if C satisfies one of the following conditions:

1. A(C)?4-(21/3-rr)r2=4-0.322509...r2, 2.

A(C)?6R2arctanR2

=4-

27R4 +

= 4 - 0.592 592... R-4+ - ...

LATTICE POINTS

22

The first condition was obtained earlier by van der Corput and Davenport (1946) under slightly more restrictive assumptions for C.

All results of this subsection call for extension to d , 2 and for generalizations in the sense of van der Corput's generalization of the fundamental theorem considered in section 3.4.

3.6 In the following we will state for d ? 2 several results about the points of the integer lattice Zd contained in a convex body. These results form a small new branch of the geometry of numbers. Note that

we do not require that the convex bodies are symmetric or have a particular position in space.

A first theorem of this type is due to Bender (1962) (d = 2), Wills (1968, 1970) (d = 3, 4) and Hadwiger (1970) (general d). Let C be a convex body in Ed for which V(C) ? S(C)/2. Then C contains at least one point of Z d. This result is the best possible. If C is a large flat disk between two lattice planes of height almost 1, then V(C) is almost S(C)/2, but we still have that C n Zd = 0. We next consider the problem of specifying lower and upper estimates

for the number of points of Z d contained in a convex body. For the estimates we use quantities of the body which do not define it precisely.

Also the particular position in space is not taken into account. Hence one cannot expect the estimates to be very precise. Using the result of Hadwiger (1970) stated before, Hammer (1971) proved the following. Suppose that a convex body C satisfies V(C)

2kS(C) for some k E {-1, 0, 1, 2, ...}. Then C contains at

least d(2k+i - 1) + 1 points of Zd. A different generalization of Hadwiger's (1970) theorem can be stated as follows. Let C be a convex body

such that the smallest integer, say n, which is greater than V(C) 'S(C), is positive. Then C contains at least n points of Zd. The non-trivial proof of this estimate was achieved through the work of Nosarzewska (1948) (d = 2), Schmidt (1972) and Bokowski and Wills (1974) (d = 3) and - for general d - of Bokowski, Hadwiger and Wills

(1972). For Nosarzewska's result and a generalization of it due to Hadwiger and Wills (1976) see section 2.2.

The lower estimates for the number of points of Z d contained in a convex body stated above are rather satisfying. Considering upper estimates the situation is much worse in spite of many efforts. The best we can offer are conjectures. Let K be the cube {x: Ixil - b (2) was solved by Sawyer (1976), see also Scott

problem for

denotes the euclidean norm. For a covering set of (1978(b)). Here minimum perimeter see Reich (1973). I

Several papers are dedicated to particular convex bodies such as triangles and parallelograms - see e.g. Hsieh (1969), Maier (1969) and

LATTICE POINTS

24

Table 3.1 Sufficient conditions for a plane convex body to contain k points of Z2 Condition

Value of k

Author

A>D

k>A-D

A>-kAD,A= 1.142607...

k

A(D-1)?D2/2,D?2

1

Baiada and Tripiciano (1975) Scott (1974), Hammer (1979) Scott (1983) Nosarzewska (1948), Bender (1962) (k = 1), Hammer (1964)

A > kP/2

k

(W - 1)A ? W2/2

1

(W-1)D(2+\/3)/2

Scott (1980, 1983) Scott (1979)

1

= 1.866 025.. . (W - 1)(D - 1) >- 1 (W - 1)P > 3W (W - 1)R ? W/V3

Scott (1979) Scott (1980) Scott (1980) Sallee (1969)

C of constant width, W >_ constant (> 1.545)

W >_ (2 + \/3)/2 = 1/866025.

Scott (1973)

W>a

Elkington and Hammer (1976)

.a]2 a

2

C of constant width,

W?a C contains o in its interior,

(W-V2)(D-2)--2

1.5461

= [0.646 830. 2

.a]2

Elkington and Hammer (1976) Scott (1985)

Niven and Zuckerman (1967).

Finally, we mention Steinhaus' problem: does there exist a point set such that no matter how it is placed on the plane it covers exactly one point of Z2? Beck (1984) has the following partial answer: there is no bounded and Lebesgue measurable set satisfying Steinhaus' property.

4

Blichfeldt's theorem

4.1

A generalization of Minkowski's lattice point theorem, different from the results of section 3, is the theorem of Blichfeldt (1914), proved independently by Scherrer (1922). Let M be a measurable set and L a

lattice in Ed. If V(M > d(L) or V(M) = d(L) and M is compact, then M contains two distinct points p, q with p - q e L. If M is compact and Jordan measurable and does not give rise to a lattice tiling (see section

10.4), then there is a constant a < 1 such that V(M) , ad(L) still yields the conclusion of the theorem. Scherrer's proof makes use of Dirichlet's box principle, also called the

pigeonhole principle: if one has to put n + 1 objects into n boxes one must put at least two of the objects into the same box. This seemingly trivial proposition has applications in Diophantine approximation - see e.g. Minkowski (1907). We can formulate Blichfeldt's theorem in the following way. Suppose

M, L fulfil the assumptions of the theorem. Then there is a translate of

M, say M + t, which contains at least two distinct points of L. (See figure 4.1.) (Sawyer (1962) and Hammer (1968) obtain analogous results by applying rotation instead of translation.)

Like Minkowski's fundamental theorem, Blichfeldt's theorem also permits a simple geometric explanation. Consider the system of translates

{M+r:reL}. If V(M) > d(L), the `total volume' of these tranlates `exceeds' the `total volume' of Ed. Hence there must be overlappings. Assume for example that (M + r) r (M + s) * 0 for suitable r, s E L with r = s. Then

p + r = q + s for suitable p, q E M and thus p - q= s- r E L\{o}.

Using the concept of density the heuristic argument in this `proof' can be given a precise meaning. 25

LATTICE POINTS

26

O

O

O

Figure 4.1 Blichfeldt's theorem. V(M) , d(L)

4.2 The fundamental theorem is an easy consequence of Blichfeldt's theorem. Let C be an o-symmetric convex body and L a lattice such that V(C) % 2dd(L). Then V(ZC) % d(L). The body ZC is compact. Blichfeldt's theorem thus implies that there are distinct points p, q E Z C

with p - q E L. Since ZC is symmetric about o we have -q E 2C and thus p - q E 2 C + 2 C= C, concluding the proof. For other applications of the theorem of Blichfeldt we refer to Gruber and Lekkerkerker (1987).

4.3

Using essentially the idea of Scherrer's proof, van der Corput

(1936) generalized Blichfeldt's theorem in the same way as he general-

ized the fundamental theorem. Let M be a measurable set and L a lattice in Ed with V(M) > kd(L) or V(M) = kd(L) and M compact, where k is a positive integer. Then M contains k + 1 distinct points p(),

,p(k+ t) such thatp 0W-pWELfor i, jE{1,

.

.

.,k+1}.

Other generalizations of the theorem of Blichfeldt in E' fall into three categories. Firstly, there are elementary refinements due to Woods

(1958(a)) and Gruber (1967(a)) based on elementary set operations and simple properties of Lebesgue measure. These refinements apply to the so-called conjecture on the product of non-homogeneous linear forms see section 12.4. The second category contains generalizations employing functions, but still in an elementary way - see Rado (1946) and Uhrin (1981). Thirdly, there are generalizations and refinements using Fourier series due to Cassels (1947) and Bombieri (1962) which are similar in spirit to Siegel's refinement of the fundamental theorem; cf. section 3.4. Bombieri applied his generalization also to the conjecture on the product of non-homogeneous linear forms.

BLICHFELDT'S THEOREM

27

In essence the proof of Blichfeldt's theorem is based on the following modern version of Dirichlet's box principle: if on a measure space of total measure 1 the integral of a real integrable function is greater than

1, then the function assumes values greater than 1. Thus the idea suggests itself to extend Blichfeldt's theorem to spaces more general than Ed, such as topological groups or homogeneous spaces. Results of this type were obtained by Tsuji (1952, 1956), Macbeath (1959) and Santalo (1955), (1976), p. 175.

4.4 An interesting adjunct result of Blichfeldt's theorem in the plane is the following result of Beck (1988) which makes use of a theorem of W. Schmidt on Diophantine approximation: there is a function

f: R + - R+, f(x) - + - as x ---> + -, such that any planar convex

body C can be placed on the plane so

as to contain at least

A(C) + f(A(C)) points of Z2. Similarly C can be placed on the plane such that it contains at most A(C) - f(A(C)) points of Z2. This result confirms a conjecture of L. Moser. Beck conjectures that the true order of magnitude of f(x) is about x1/4, as x ---> + -.

5

Successive minima

5.1 In chapters 3 and 4 we have considered the lattice point theorem of Minkowski and several generalizations and results related to them. Yet another generalization is based on the notion of successive minima. In this section we will present this generalization due to Minkowski and also some related results. There are several important applications of these results in Diophantine approximation and to the reduction theory of positive quadratic forms. Among the numerous papers dealing with applications of successive minima we mention Mahler (1966), Schmidt (1969, 1970, 1980) and Jurkat and Kratz (1981). For further references the reader is referred to Gruber and Lekkerkerker (1987). Let S be a star body and L a lattice in Ed. The homogeneous or first successive minimum A = A1(S, L) of S with respect to L is defined by Al = inf{A > 0: AS contains a point

o of L}.

Clearly L is admissible for x.1S. If S is compact, then there are points of

L contained in the boundary of S. Define the successive minima Al = A1(S, L), ..., Ad = Ad(S, L) of S with respect to L by A, = inf{A > 0: AS contains i linearly independent points of L}. (5.1) Obviously,

0,Al ,A2 5.2

...,Ad, b 2(l),

.

.

.

b ('),

.

.

.,

b 1(d), b 2(d),

.

.

.

b (d).

Let £ be endowed with the topology thus defined. 37

LATTICE POINTS

38

7.2 The original form of Mahler's selection or compactness theorem is the following: a sequence of lattices with determinants bounded above

and which are all admissible for a neighbourhood of the origin o contains a convergent subsequence. This implies in particular that £ is locally compact.

A nice proof of Mahler's theorem was given by Groemer (1971). Since it exhibits a connection with another important geometric selection

theorem, the selection theorem of Blaschke (1916), we describe the main idea of the proof. For a lattice L define the Dirichlet-Voronoi cell D(o, L) of L by

D(o, L) = {x: xI < Ix - pl for all p E L}. (For more information see section 10.4.) Let L1, L2, ... be a sequence of lattices in El with determinants bounded above and which are all admissible for some neighbourhood of o. Consider the suitably topologized space of compact convex subsets of V. In this space the sequence D(o, L1), D(o, L2), . . has a convergent subsequence by Blaschke's selection theorem. The limit turns out to be the Dirichlet-Voronoi cell of a suitable lattice L and L is the limit of the corresponding .

subsequence of lattices.

7.3 In order to present an application of the selection theorem we outline a proof that for each convex body C in Ed there is a lattice L such that the system { C + p : p E L) is a lattice packing of C which has maximum density among all lattice packings of C, where the density is V(C)/d(L) (cf. sections 3.2 and 8.1). For a lattice L the system { C + p : p E L } is a packing if and only if

L is admissible for the convex body C - C = {x - y; x, y E C}. This

body is a neighbourhood of o. Let L1, L2, ... be a sequence of lattices such that (C + p: p c- L, } is a packing of C for each i c- { 1, 2, . . .} and

such that the densities V(C)/d(L,) tend to the supremum (>0) of the densities of the lattice packings of C. The lattices L1, L2, . . are all admissible for the neighbourhood C - C of o and their determinants are bounded above. Hence Mahler's selection theorem implies the existence of a subsequence of L1, L2, . . converging to a lattice L. It can be shown that L is still admissible for C - C and thus provides a .

.

lattice packing of C. The density of this lattice packing is maximal. For other applications of the selection theorem the reader is referred to the papers of Mahler (see the references in Cassels (1972) or Gruber and Lekkerkerker (1987)) and to Gruber and Lekkerkerker (1987).

7.4

Chabauty (1950) extended Mahler's selection theorem to a uni-

MAHLER'S SELECTION THEOREM

39

formly discrete sequence of subgroups of a locally compact topological group. Some other extensions and applications can be found in Macbeath and Swierczkowski (1960), Mumford (1971) and Harvey (1974). Hammer and Dwyer (1976) and Dwyer et al. (1983) consider compactness theorems for sequences of pairs each consisting of a convex body and a lattice. In particular they deal with the case when the lattices are packing respective covering lattices of the convex bodies (see section 8.1). Another example is provided by Dirichlet-Voronoi cells and the corresponding lattices, essentially considered by Groemer in his proof of

Mahler's selection theorem. These results provide direct proofs for existence theorems like the one in section 7.3.

8

Packing and covering

8.1 The theory of packing and - to a lesser degree - the theory of covering with convex bodies or star bodies, has attracted the interest of

several generations of mathematicians. There also exists a body of articles dealing with packing problems in physics, chemistry, biology and

technology. Early references are Reynolds (1885) (see also Coxeter (1969), §22.4) and Kelvin (1904). Whereas earlier results deal mainly with packing of balls and the problems of tiling (see chapters 9 and 10),

the general theory was developed starting with Minkowski's (1904) paper on lattice packing. The investigation of a more general type of packings was suggested by Hilbert who, in the context of his 18th problem, said the following: I draw your attention to the following question related to the preceding one, and important to number theory and perhaps sometimes useful to physics and chemistry: how can one arrange most densely in space an infinite number of equal solids of given form, e.g. spheres with given radii or regular tetrahedra with given edges (or in prescribed position), that is, how can one fit them together so that the ratio of the filled to the unfilled space may be as great as possible? (Hilbert (1900).)

Important later work is due to Fejes T6th, Bambah, Rogers, Zassenhaus and many others. The monograph of L. Fejes T6th (1972), first published in 1953, spurred a great deal of current research on packing and covering.

Basic references on packing and covering are Hlawka (1949), C. A. Rogers (1964), L. Fejes T6th (1964, 1972, 1984), Baranovskii (1969), G. Fejes T6th (1983) and Florian (1987).

A family of subsets of E' is a packing if any two of these sets have distinct interiors. We will consider mainly packings of congruent copies or of translates of a given convex body or a compact star body C. If a packing of C is of the form {C + p: p E L}, where L is a lattice, it will 40

PACKING AND COVERING

41

be called a lattice packing of C with packing lattice L. Call a family of subsets of Ed a covering if its union equals Ed. It is clear what is meant by covering with congruent copies or translates, by lattice covering and by covering lattice.

In order to avoid difficulties we define the concept of density only for packings and coverings with sets having uniformly bounded diameters. If {Ck: k E I} is such a packing, its density is defined as

Ck n {X: 14

lim supp

Q} * 0}

(2a) d

In case {Ck: k E I} is a covering, define its density by E{V(Ck): Ck C {X: Xij < Q}} lim inf

(20) d

a-_+.

The density of a packing can be interpreted as the `proportion of space' covered by the sets of the packing or as the probability that a randomly chosen point of Ed is contained in a set of the packing. The density of a covering is - cum grano salis - the `total volume' of all sets `divided' by the `total volume' of the whole space. Let C be a

convex body or a compact star body. If {C + p: p E L} is a lattice packing or a lattice covering, the density is V(C)/d(L) in each case. Let 5L(C), OT(C) and bc(C) denote the (attained!) suprema of the densities of the lattice packings of C, of the packings of translates of C and of

the packings of congruent copies of C, respectively. For coverings define similarly OL(C), OT(C), Oc(C) where the suprema are replaced by the infima. 5L(C) and OL(C) are called the densities of the densest lattice packing and of the thinnest lattice covering of C, respectively, and correspondingly for bT(C), bc(C) and OT(C), Oc(C). Obviously

OL(C) < 640 < MO

1 < OC(C)

OT(C) < 0J0-

(8.1)

For the first and last inequalities, it is not known whether there are convex bodies C for which the inequalities are strict. See the discussions in sections 8.3, 8.4 and 9.2. There are star bodies C for which the first and last inequality are strict, see section 8.4.

For a convex body C the problem of finding 6L(C) is equivalent to

the determination of the critical determinant 0(C - C) of the osymmetric convex body C - C. This can be seen from the formula 6L(C) =

V(C)

0(C - C)' A family of subsets of Ed is a k -fold packing or a packing of multiplicity k, k E {1, 2, ...}, if each point of Ed is an interior point of at most k of the sets. The multiplicity is precisely k if it is k and if some

LATTICE POINTS

42

point in E' is an interior point of k of the sets. Call a family of subsets of El a k-fold covering or a covering of multiplicity k if each point of Ed is contained in at least k of the sets. We say that the multiplicity is precisely k if it is k and some point of Ed is contained in precisely k of the sets. All concepts defined in the last two paragraphs can easily be transferred to this more general situation. Thus in particular we may , 9'(C). consider the quantities 8'(C), There is no exact duality between packing and covering but we will arrange the material in such a way that similar results are treated in the same subsections. This has the advantage of showing that progress is not uniform for packing and covering.

8.2

This subsection contains estimates for the quantities SA(C), ST(C) and OL(C), OT(C), where C is a given convex body in Ed. For the cases d = 2 and d = 3 we refer to 8.3. If C is symmetric in o, then the refinement of the Minkowski-Hlawka theorem due to Schmidt (1963) (see (6.6) in section 6.2) yields add < 5T(C) where ad -> logV2 = 0.346573 ... as d ---> +-. 2d-1 (8.2)

We should like to point out the large gap between the lower and upper bound for 8T(C) in (8.2) and (8.1). At present it seems to be difficult to tell whether (8.2) can be improved essentially. Our conjecture is that it probably cannot. Compare also the discussion for balls in section 9.3. Assume now that C is not necessarily symmetric. If { C + p ('), i E I) is a packing of translates of C, then an argument of Minkowski (1904), §1, shows that { 1(C - C) + p ('), i E I) is also a packing and vice versa. Hence SG(C) =

ST(C) =

V (C)

V(1z(C - C)) V(C) V(21

(C - C))

SL(z(C - C)), 8T(2(C - C)).

(8.3)

In a surprising recent work, Rush (198?) proved the following. Let C be a convex body which is symmetric through each of the coordinate hyperplanes. Then there is an explicit construction of a lattice packing of C of density 2-d + o(d)

as d ---> +

(for infinitely many d's).

This construction is based on a particular code and makes use of the

PACKING AND COVERING

43

contructions of Leech and Sloane (1971) for bal, packings, see section 9.4.

By inequalities of Rogers and Shephard (1957) (see also C. A. Rogers (1964), ch. 2) and Brunn and Minkowski (see, for example, Bonnesen and Fenchel (1934), §11) we have 2d V(C) < c 1. C)) V(((2 (C

-

))

Here equality holds in the first inequality if and only if C is a simplex and in the second inequality if and only if C is centrally symmetric. Together with (8.2) and (8.3) this shows that 2add where ad - log '\/2 = 0.346573 . . as d - +oo. (2d) C SL(C) Note that d3/2 2add -\/rrlog V2d3/Z = 0.614285... 4d as d (2d) 4d .

d 1

Thus for the non-symmetric case the lower estimates are even worse than the already very weak ones for the symmetric case.

For covering the situation is much better. While earlier attempts produced for OL(C) upper bounds of the form ad for suitable a > 1, C. A. Rogers (1957, 1959) succeeded in proving that for a convex body C

OT(C) _ 3, we have SL(C) = 6T(C). Most probably the answer for d = 3 is yes but there are reasons to conjecture that for d as small as 10 or 11 it is no - see section 9.2. It is an open problem for what centrally symmetric plane convex bodies C does the value of SL(C) attain its minimum. Surprisingly the minimum is not attained for circular discs. Work of Reinhardt (1934) and Mahler (1947) supports the conjecture that the minimum is attained for regular octagons which are suitably smoothed at the vertices. Thus the conjecture says that for any centrally symmetric convex body C in the plane the following inequality holds:

6L(C)>

8-4\/2-log2 = 0.902414 2V2-1

A result of Tammela (1970) which improves upon an earlier theorem of Ennola (1961) shows that for all centrally symmetric convex bodies C in E2,

The bound in Tammela's result is surprisingly close to the conjectured optimal bound.

PACKING AND COVERING

45

For a convex body C in E2 that is not necessarily symmetric a result of Fary (1950) says that

6L(C) - 3', where equality holds for triangles only. (See figure 8.1.)

(a) SL(T) = ST(T) = 2/3

\Z\Z\/\/\/ VVV (b) bc(T) = 1

Figure 8.1 Most dense packing of translates and of congruent copies, respectively, of a

regular triangle T

The problem of determining the density of the densest lattice packing of a particular convex body or star body in general is a difficult task. There is no effective machinery available to tackle this problem. Table 8.1 contains an (incomplete) list of convex bodies for which the densities of the densest lattice packings have been determined. The densities are given only in the cases where they can be expressed in a simple manner. For further results consult Keller (1954) and Gruber and Lekkerkerker (1987). A recent result of Kuperberg (1987(a)) says that for a plane convex

body C we have

bc(C) ,

2s 32

Now we turn to coverings. Let C be a plane convex body and denote

by K (resp. T) a convex hexagon (resp. triangle) of maximum area inscribed into C. A general conjecture asserts that

Oc(C) , A(K). This has not yet been proved, but L. Fejes Toth (1972), p. 86, gave a

result which comes close to it. Fejes Toth's result implies that OT(C) -- A(K).

(8.9)

4

{x: ((x1)2 + (x2)2)1/2 + Ix31 < 1}

{x: IXI !5 1,x31 - Al

Ball

Cylinder with convex base C'

1x: IX1 + X2 + x31 Ixl xtrTtrATx

or in terms of the coefficients d

aik -

tlitmkalm

for (a ii,

.

.

.)tr

c-

Ed(d+l)/2.

l,m=l

We have thus obtained a linear transformation .7 of Ed(d+1)/2 onto itself. Clearly = (-/).

11.2 When are two given (positive quadratic) forms (on E d) equivalent, or, geometrically, when do two lattices (in Ed) differ by a rotation only? This seemingly simple question constitutes one of the origins of reduction theory. It leads to the following more precise questions. As is customary we formulate them in terms of forms rather than for bases of lattices.

GEOMETRY OF POSITIVE QUADRATIC FORMS

83

1. Specify a set (Q of forms in (7) such that each form is equivalent to (a) one, or (b) a bounded number of forms of (Q. We call (Q a reduction domain and the forms of (Q reduced forms. In the cases considered in

the literature (Q is in general a convex polyhedral subcone of () with apex at the origin and with parts of the faces possibly deleted. (In many cases one does not care about the boundary of (Q.) Sometimes (Q is a finite union of such cones. Each form in () is equivalent to a form in (Q.

For an interior form, i.e. a form contained in the interior of (Q, we require either that it is not equivalent to any other reduced form or that it

is equivalent to a fixed number, say k, of reduced forms. In the

former case (Q is a simple reduction domain, whereas in the latter case (Q is called a multiple reduction domain of multiplicity k. (A reduced form on the boundary of (Q may be equivalent to more than one form in (Q even if (kis a simple reduction domain, and to a number (different from the multiplicity) of forms in (-if (Q is a multiple reduction domain.) By a tiling of (7 we understand, by analogy to the definition in section 10.4, a family of subsets of 7 whose union equals 7 and such

that any two distinct sets have disjoint interiors. If (Q is a simple reduction domain, then T an integer unimodular matrix}

is a tiling of (P. If (Q is a convex polyhedral cone or a finite union of such then a (not necessarily positive) form on an edge of (Q (which may be contained in the boundary of ()) is an edge form. 2. Specify an algorithm by means of which one can transform a given form into an equivalent reduced form.

3. Given a reduced form, what are the integer unimodular matrices by means of which one can transform it into all equivalent reduced forms?

There exist several different definitions of reduction due to Lagrange (1773), Seeber (1831), Gauss (1831), Dirichlet (1850); Hermite (1850); Korkin and Zolotarev (1872, 1873, 1877); Selling (1873/74), Charve (1882); Minkowski (1905); Hofreiter (1933(a)); Venkov (1940, 1952) and

others. For detailed information on reduction we refer to Delone (1937/38), Keller (1954), van der Waerden (1956), van der Waerden and Gross (1968), Ryskov (1980), Lenstra, Lenstra and Lovasz (1982) and Gruber and Lekkerkerker (1987). In the following we will study Minkowski reduction only. A form q(x) = I a;1x,xk is reduced (in the sense of Minkowski (1905)) if for each i E {1, ..., d} the inequality

q(u,,...,ud)aii =q(0,...,0,1,0,...,0) ith position

(11.1)

LATTICE POINTS

84

holds for any u = (u 1, ... , u d) cr E Z d for which the greatest common divisor of ui, ..., Ud is 1 and if for each i E {1, ..., d - 1} we have ai,r+i > 0.

(11.2)

(Note that it is not required that q is positive.) In more recent literature the conditions (11.2) are omitted in general. If this is the case then one speaks (somewhat unexpectedly) of classical

Minkowski reduction. In order to clarify the geometric meaning of Minkowski reduction for positive forms we state a geometric version of the definition: a basis {b('), ..., b(d)) of a lattice L in Ed is reduced (in the sense of Minkowski) if

b(') E {b E L\{o}: jbI is minimal),

b('+') E {b E L\{o}: 0),

...,

b''), b can be extended to a

basis of L, I b I is minimal), b(1)) > 0, (b('+l)

for i E {1, . ., d - 11. In the following we will use the first version of the definition only. .

If q is a positive reduced form, then

all = min{q(u): u E Z'\{o}). It was shown by Minkowski (1905) that for any positive form there is an equivalent reduced form. Minkowski (1905) proved two finiteness theorems. The first finiteness

theorem can be stated in the following way: There is a finite set Ul u u Ud C Zd such that a (not necessarily positive) form q satisfies all inequalities (11.1) if and only if it satisfies the finitely many inequalities

q(UI,...,ud)?aii

for(ul,...,ud)'E Ui, i E {1.... ,d}. (11.3)

Hence q is reduced if and only if it satisfies the inequalities (11.2) and (11.3). Minimal sets of such inequalities (11.3) were given by Lagrange (1773) (d = 2), Seeber (1831) and Gauss (1831) (d = 3) and Minkowski (1883) (d = 4). The set of inequalities specified by Minkowski (1887) for d = 5, 6 is sufficient as proved by Ryskov (1973) and Afflerbach (1982)

(d = 5) and Tammela (1973) (d = 6), but not minimal. A minimal subset of Minkowski's conditions for d = 5, 6 is due to Tammela (1973(a)). The case d = 7 was treated by Tammela (1977). For d = 8 minimal sets of such inequalities are not yet known. We state the results for d _ 1. Each facet of hyperplane of the form d

{(till,

V12,

.

.

., Vdd) tr

c-

Ed(d+1)/2: E UikUiUk = 1} i,k=1

(vik = Vki),

where u = (ul, ..., Ud) tr E Zd\{o} and U1, 2.

. ., Ud are relatively prime. The forms in the facet have arithmetic minimum 1 and attain it at u. Each ray in /-) starting at the origin intersects the boundary of J/1 in .

precisely one point. The discriminants of the forms in J/t are bounded below by a positive constant.

To any integer unimodular d x d-matrix T we assign a linear transformation 7 of E d(d+1)/2 according to section 11.1. Two sets in E d(d+1)12

are called equivalent if there is a transformation 7 mapping one of the sets onto the other one. 3.

J/t is invariant with respect to the group of all such transformation ,7. Any two facets of _//t are equivalent and there are only finitely many vertices of J/1 which are pairwise non-equivalent. Such a set of vertices can be found by the so-called algorithm of Voronoi (1908).

The (equi-)discriminant surface "/) = (/)(a) (a > 0) consists of all forms in O of discriminant a. It is a strictly convex smooth surface and each ray in (7) starting at the origin meets it exactly once. Using the concepts of J/2 and '/) Ryskov (1970) easily proved the following theorem: a form q on the boundary of J/1 is extreme if and only if it is a vertex and if the discriminant surface (/) through q contains

a suitable neighbourhood of q in J/t. The latter condition can be expressed by saying that the tangent hyperplane of (/) at q intersects J/1 at q only. (It is interesting to note that this deep theorem, and thus also Voronoi's theorem, which will be stated below, has an almost trivial

proof. This phenomenon is also encountered in Minkowski's fun-

damental theorem, in Blichfeldt's theorem and in some sense also in Mahler's compactness theorem. The latter is highly plausible; the difficulty in the proof is to find the right technical tool.) We draw some consequences of Ryskov's theorem: A vertex of J/1 is contained in at least d(d + 1)/2 facets of J/t. This implies that an extreme form q attains its arithmetic minimum in at least d(d + 1)/2 points of Zd\{o} (Korkin and Zolotarev (1872)). Since q is the intersection of the hyperplanes containing the facets in which q is contained, it is uniquely

GEOMETRY OF POSITIVE QUADRATIC FORMS

91

defined by its minimum points. In other words, q is perfect. The tangent hyperplane of (/) through a vertex (aik) of Al has the equation d

I bikVik = d

(Vik = Vki),

1,k=l

where (bik) _ (aik)-1. Using a well-known result from convexity we have that if the tangent hyperplane intersects J/l at (aik) only, its normal vector is a positive linear combination of the exterior normal vectors of the facets of Al containing (aik). Thus if 00, ..., u(^') E Zd\{o} are the points at which the form q(x) = Edk=1 aikxiXk attains its arithmetic minimum, then (b11, 2b12, .

.

., 2bd-ld, bdd)t

is a linear combination with positive coefficients of ((u(,'))', 2U1(1)ll2(1) ,

(1) (I) (1) ) 2 tr ., 2ud-lud , (ud ) ,

(N) (n) 2u1(N)u2(N) , . ., 2ud-lud , (ud(N)) 2)tr In other words, an extreme form is eutactic. Conversely it is easy to (((N)

ul

)

2

,

.

show that a perfect and eutactic form is extreme. We thus obtain the following fundamental theorem of Voronoi (1908(a)): a form is extreme

if and only if it is perfect and eutactic. Using this result, one can determine in principle whether a given positive quadratic form is extreme, or equivalently whether a given lattice provides a ball packing

of locally maximum density. For the Leech lattice this was done by Gruber et al. (1987) - see section 9.6.

In order to find all extreme forms it

is

sufficient to consider a

maximal set of inequivalent vertices of J/2 and to take those vertices q for which the tangent hyperplane of the discriminant surface through q intersects J/I at q only. (Compare property (3) of J/l.) In dimensions d = 2, 3, 4, 5, 6 there are (up to multiplication with a positive factor and up to equivalence) 1, 1, 2, 3 and 6 extreme forms respectively. For d s 5 the extreme forms were determined by Korkin and Zolotarev (1872, 1873, 1877). Their result was verified by Voronoi (1908(a)). Barnes (1957) determined the extreme forms for d = 6. For lists of certain extreme forms in higher dimensions and many results we refer to Ryskov and Baranovskii (1979) and Gruber and Lekkerkerker (1987). On a computer, the determination of all extreme forms even for

relatively modest dimensions at present seems to be out of reach, because of the huge amount of time and space required.

LATTICE POINTS

92

For a remark on the relation of locally densest lattice packings of balls and local minima of the zeta-function for lattices see the last paragraph in section 15.2.

We next consider the connection of the Minkowski reduction domain

iQ+ and J/l. For each facet of Jt consider the cone with apex at the origin generated by this facet. This gives a tiling of (P. A different tiling of (/) is obtained by considering the distinct 7/Q+ among the cones where T is an integer unimodular matrix. It is not difficult to show that the tiling derived from (Q+ is a refinement of the tiling connected with

A. From this one may deduce that each perfect form, i.e. a form contained on a ray from the origin through a vertex of J/1, is equivalent to a so-called edge form but not every edge form is perfect. An edge form (or Kantenform in German) is a form on a 1-dimensional edge of It follows that each extreme form is equivalent to an edge form (Q+.

(Minkowski (1905)).

11.4 Voronoi (1908(a)) did not use the polyhedron -/#. This was introduced by Ryskov in 1970. Instead he used the so-called Voronoi polyhedron II(d). It is defined as the convex hull of all points in E d(d+1)/2 of the form

/

z \u1,

ulu2,

, ulud,

z

uz,

., ud-lud, Ud) cr, z

(11.5)

where (u1, ..., ud)1` E Zd\{o} has the property that u1, ..., Ud have greatest common divisor 1. Note that the points of the form (11.5) are on the boundary of (P. One can prove that II(d) = {A E Ed(d+1)/2:

1 for all V E Jn),

where is the inner product in Ed(d+1)/2Thus II(d) is `dual' to Jn and each property of Al is reflected in a corresponding property of 1I(d).

We list some properties of II(d). 1.

rI(d) is a convex polyhedron (in a more general sense) in the

closure of (P. Each face of lI which is not contained in the 2. 3.

boundary of is a (compact) polytope (with finitely many vertices). The unbounded faces of 11 are of dimension -- d(d - 1)/2. Each ray in 7) starting at the origin intersects the boundary of II in exactly one point. II(d) is invariant with respect to the group of all transformations 7

GEOMETRY OF POSITIVE QUADRATIC FORMS

93

where T is an integer unimodular matrix. There are only finitely many faces of I7(d) which are pairwise non-equivalent.

The determination of the space-group types and of the Bravais types (see section 10.3) requires the knowledge of a representative from each class of integrally conjugate finite groups of integral unimodular d x dmatrices. Here two such groups are integrally conjugate or integrally equivalent if there is on isomorphism of the form

T-> S-'TS with a fixed integer unimodular matrix S mapping the first group onto the second.

A basic theorem of Jordan (1880) says that there are only finitely many classes of integrally conjugate finite groups of integral unimodular d x d-matrices. Such a class of group is also called an arithmetic crystal

class. For d = 2 there are 13 arithmetic crystal classes, for d = 3 there are 73 (see Delone (1934(a)), Neubuser and Wondratschek (1969) and Brown et al. (1978), and for d = 4 there' are 710 (see Dade (1965), Bulow (1967), Billow and Neubuser (1970) and Brown et al. (1978)). A

theorem of Maschke shows that for each finite group G of integer unimodular matrices there is a form q such that G is a subgroup of the

group of automorphisms of q, i.e. the group of integer unimodular matrices T such that

q(x) = q(Tx)

for all x E Ed.

Thus the problem of determining the arithmetic crystal classes reduces to the determination of the (integral conjugacy classes of) groups of

automorphisms of all forms. A result of Ryskov (1972(a), 1972(b), 1972(c)) says that these groups are among the groups of automorphisms of those forms which are the centroids of a maximal system of pairwise non-equivalent faces (of all dimensions) of 17(d) (or of Jit). Jordan's theorem is an immediate consequence of this. Thus using the algorithm of Voronoi (1908(a)) it is possible in principle to determine all arithmetic crystal classes in any dimension.

11.5 We next consider the problem of locally thinnest lattice coverings with balls. For reasons of simplicity we use lattices rather than forms. We will follow the papers of Delone, Dolbilin, Ryskov and Stogrin (1970), Delone and Ryskov (1971) and Ryskov and Baranovskii (1976). The main tools are L-tilings and L-types.

Let L be a lattice in Ed. Denote by a(L) the minimum a > 0 such that L is a covering lattice of aBd. Call a(L) the covering radius of L.

LATTICE POINTS

94

meant by the density of the lattice covering {a(L)Bd + p: p E L} and by a locally thinnest lattice covering with balls (see sections 8.1 and 11.3). The empty ball method of Delone (1928, 1934(a)), by means of which L-tilings are introduced, can be described as follows: a ball in Ed is an empty ball for the lattice L if it contains no point of L in its interior. If It is clear what is

an empty ball contains a system of non-coplanar points of L on its boundary the convex hull of these points is an L-polytope. The system of all L-polytopes forms a facet-to-facet tiling of Ed, the L-tiling or L-partition or Voronoi partition of L. It is also called Delone triangulation or partition. See figure 11.2. This concept is dual to the DirichletVoronoi tiling of L as introduced in section 10.4: An L-polytope is the convex hull of the centres of a system of all Dirichlet-Voronoi cells of L having a common vertex. It is easy to see that a(L) equals the maximum radius of a circumsphere of an L-polytope. L-tilings were first introduced and studied by Voronoi (1908(a), 1908(b), 1909). Later on they were intensively studied by Delone and his school.

Figure 11.2 L-tiling

We now describe the concept of L-type due to Voronoi. Two lattices

L and M are said to belong to the same L-type if there is a linear transformation of Ed which maps L onto M and at the same time the L-tiling of L onto the L-tiling of M. Positive quadratic forms q and r are of the same L-type if they are metric forms of lattices belonging to the same L-type. A lattice L (or a form q) belongs to a primitive L-type if the L-tiling of L (of a lattice of which q is the metric form) consists

GEOMETRY OF POSITIVE QUADRATIC FORMS

95

of simplices only. In dimension 2 there are one primitive and one non-primitive L-type (Fedorov (1885)); in dimension 3 there are one primitive and four non-primitive L-types (Voronoi (1908(a), 1909) and in dimension 4 there are three primitive and at least 49 non-primitive L-types (Delone (1929), Stogrin (1973)). The 221 primitive L-types for d = 5 were determined by Ryskov and Baranovskii (1976). In principle

- but only in principle - it is possible to determine all finitely many (primitive) L-types in any dimension - see Voronoi (1908(a), 1909) and Ryskov and Baranovskii (1976).

The primitive L-types of positive quadratic forms - considered as subsets of q) - are unions of disjoint open polyhedral convex cones with

apices at the origin. The non-primitive types are contained in the boundaries of these cones. The closures in (P of these cones form a tiling of (P.

A remarkable result of Barnes and Dickson (1967(a)) says the following: in any cone of this tiling of q) there is (up to multiplications with a positive factor) at most one form q such that a lattice with metric form q provides a locally thinnest lattice covering with balls. If T is an integer unimodular matrix, such that the cone which contains such a form q is invariant with respect to 9, then T is an automorphism of q. Delone, Dolbilin, Ryskov and Stogrin (1970) and Delone and Ryskov (1971) make the result of Barnes and Dickson even more transparent by giving a very clear geometric proof of it. In dimensions 2 and 3 the thinnest lattice coverings thus provide the only locally thinnest lattice coverings whith balls. In dimension 4 there are three locally thinnest lattice coverings with balls (Baranovskii (1965(a), 1966), Barnes and Dickson (1967(b)).

12

Selected problems of number theory

12.1 Geometry of numbers and lattice point problems of analytic number theory are branches of number theory in which the notion of lattice plays a central role. Several other parts of number theory either make use of results on lattices or may be interpreted in terms of lattice points. Examples are quadratic forms, Diophantine approximation and Diophantine equations. Continued fractions are related to lattice point results as well and the even more distant algebraic number theory still has connections with lattice point methods.

In this section we will first describe the so-called parallelogram algorithm and relate it to Mordell's converse problem for the linear form theorem of section 3.3, to a geometric theory of continued fractions which goes back to Smith, Klein and Minkowski and to the conjecture on the product of non-homogeneous linear forms. Then we touch on Diophantine equations and finally we will consider lattice point problems for `large' convex bodies.

12.2

In this subsection we will present a method, the so-called

parallelogram algorithm, which assigns to each planar lattice a sequence of rectangles. This algorithm and similar ones have been in use since the times of Klein (1895, 1895/96, 1897), Minkowski (1896) and Delone (1947). For historical remarks see Keller (1954). For applications see the subsequent sections. Unless stated otherwise we consider the planar case

only. A rectangle will always have centre at the origin o and edges parallel to the coordinate axes. Following Minkowski (1896), §45, we will describe the parallelogram

algorithm. Let L be a lattice in E2. A rectangle R is called extremal with respect to L if L is admissible for R and if L contains relative interior points of each edge of R. Extremal rectangles clearly exist.

Consider one of them. If no point of L on its horizontal edges is 96

SELECTED PROBLEMS OF NUMBER THEORY

97

situated on the x2-axis one may obtain a further extremal rectangle as follows: Contract the given extremal rectangle in the direction of the xl-axis until we obtain a rectangle such that the only points of L on its boundary are vertices. Then expand this rectangle in the direction of the x2-axis until the horizontal edges are `stopped' by a point of L. Thus we have obtained a second extremal rectangle. Clearly, one can make an

analogous construction with the roles of the horizontal and vertical edges, respectively the x2-axis and the xl-axis interchanged. Continuing this process in both directions we get a sequence of extremal rectangles

., R-1, Ro, R1, . . which may be finite or infinite in each of the two directions. .

.

.

(12.1)

Let (rn, sn), sn , 0, be the point of L in the relative interior of a vertical edge of R. (This is unique except for the case when Rn is the first or the last rectangle in the sequence (12.1). In these cases choose (rn, sn) respectively such that rnrn+l < 0 or rnrn_1 < 0). We obtain a sequence of points of L (12.2) ., (r-1, s-1), (ro, so), (r1, s1), ... lying alternately in the first and second quadrant. By renumbering (12.1), (12.2) if necessary, we may suppose that the points with even indices are all contained in the first quadrant. The construction of the sequences (12.1), (12.2) is called the parallelogram algorithm (figure .

.

12.1).

The following properties of the sequences (12.1), (12.2) are immediate: 1.

2.

Each extremal rectangle belongs to the sequence (12.1). We have (rn, SO = (rn-2, Sn-2) + an(rn-1, Sn-1) for n = 0, ±1, ±2, .

.

.

with suitable positive integers an. The point (rn, sn) is the last point of L on the ray starting at (rn_2i sn_2) and having direction (rn_1i sn_1) which is contained in the same quadrant as (rn-2, sn-2). 3. d(L) _ (-1)n (rnsn-1 - rn-1sn) for n = 0, ±1, .. . 4. A(Rn) = 4(-1)nrnsn+l for n = 0, ±1, ..., where A(Rn) denotes the area of R.

There exist several higher-dimensional versions of the parallelogram algorithm, see e.g. Minkowski (1907), §IV, and Keller (1954), §33.

12.3

This section contains several applications of the parallelogram algorithm. Before reading the first application the reader is advised to

I

RZ

(r2,s,)

R,

Figure 12.1 The parallelogram algorithm

(r3,s3)

Ro

(ro,so)

SELECTED PROBLEMS OF NUMBER THEORY

99

take a second look at section 3.3, where we considered Mordell's converse problem for the linear form theorem of Minkowski. In order to

be able to treat the converse problem in the planar case in a more geometric way we define for L a lattice in E2: TI T2

K(L) =sup

d (L)

: L admissible for {x: Ixi

Ti, i E {1, 2}}}

= sup{ A(R) : R extremal with respect to L}.

4d(L)

Using properties (1)-(4), Suranyi (1960/61, 1971) elegantly proved that for each plane lattice L

K(L) ,

2+

2

= 0.723 606... = K(2)

(12.3)

1

where K(2) is defined in section 3.3. Here equality holds precisely for lattices of the form DLO, where D is a diagonal transformation, and

_V5-1 1 1

2

_5+1 2

is a basis for Lo. (Lo is admissible for the star body {x: IXIX21 :!S; 1) and

has minimal determinant among all such lattices, i.e. is a critical lattice of this star body.) For lattices L with K(L) > K(2) the stronger inequality

K(L),2+21 =0.788675...

(12.4)

holds, equality being attained precisely for lattices of the form DL

1,

where D is a diagonal transformation and the lattice L 1 has basis

V3-1 (1)'

2

V3+1 2

Using the connection between the sequence (12.2), the numbers a in property (2) and continued fractions, Suranyi (1971) gave another proof

of the theorem of Szekeres (1936(b)) that, in the sense of Lebesgue measure in E4, for almost all bases {b(1), b(2)} (considered as points in E4) we have for the corresponding lattice L the equality K(L) = 1 - see

LATTICE POINTS

100

also Gruber (1971).

The phenomenon that the lattices for which (12.3) holds with strict inequality even fulfil (12.4) is an example of a so-called isolation theorem. Isolation occurs frequently in Diophantine approximation and geometry of numbers - see, for example, Cassels (1957). The close relation between the parallelogram algorithm and regular continued fractions was pointed out by Minkowski (1896), §45, who used this relation to prove some basic results on continued fractions. Instead of giving details we present a geometric interpretation of regular continued fractions due to Klein (1895, 1895/96, 1897).

Let a > 0 be irrational. We assume that the ray

S = {x: x2=ax1ix1>0) takes the role of the positive x2-axis, and instead of L we consider the integer lattice Z2 for the parallelogram algorithm. Then the points (0, 1), (1, [a]) appear in the sequence corresponding to (12.2). Introducing a new notation we consider the (unbounded) subsequences (12.5)

(q -i, p-i) _ (0, 1), (q1, p1), (q3, P3), ...

(12.6) (qo, Po) _ (1, [a]), (q2, P2), (q4, P4), ... on the left- and on the right-hand side of S. The (unbounded) polygonal lines with vertices in (12.5) or (12.6) are called Klein polygons (see

figure 12.2). The Klein polygons form part of the boundary of the convex hull of the points of Z2 in the (closed) first quadrant which are above (or below) the ray S, excluding o. Properties (2), (3) translate into

(qn, p,) = (qn-2, Pn-2) + an(gn-1, Pn-1) for n = 1, 2,

2'.

.

.

.

with

suitable positive integers al, a2, . . . The point (qn, pn) is the last point of Z2 on the ray starting at (qn-2i Pn-2) in direction (qn-1, Pn-1) before the ray intersects S.

gnPn-1 - q.-lpn = (-1)n for n = 1, 2, .. Using (2'), (3') and the geometric interpretation, one easily deduces 3'.

.

that

Po

P2

qo

q2

l

a-

< ... < a < ...
1 contains at

most finitely many rational points. Using deep tools from algebraic geometry, Mordell's conjecture was proved by Faltings (1983, 1984).

12.6 A classical number-theoretic problem which goes back to the circle problem of Gauss is to estimate the number of points of Z" contained in or on the boundary of large balls or ellipsoids, Important contributions to this difficult problem were given by Landau, Walfisz, van der Corput, Jarnik and others. In more recent times other sets besides ellipsoids have been considered. For surveys covering this more

general situation see Fricker (1982) and Gruber and Lekkerkerker (1987). In the following some results concerning convex bodies or more general sets are cited.

SELECTED PROBLEMS OF NUMBER THEORY

105

Let C be a convex body which contains o in its interior. A result of Hlawka (1950(b)) says that if C is sufficiently smooth and has positive gaussian curvature, then AdV(C) - L(AC) = O(),f)

(12.9)

as A ---> +

where /3 = d(d - 1)/(d + 1). (Recall that L(AC) is the number of points

of Zd in W. The number of points of Zd on the boundary of AC is denoted by L*(AC).) Hlawka also showed that /3 may not be replaced by

(d - 1)/2. There exist several alternative proofs and refinements of these results. An extension of Hlawka's estimate (12.9) to smooth compact o-symmetric star bodies is due to Berard (1978). Let C be a compact plane set bounded by a Jordan curve of class C?°° such that at any zero of the curvature the curvature vanishes at most of order k. A result of Colin de Verdiere (1977) says that A2A(C) - L(AC)= O(A2/3) A2A(C) - L(AC)= O(Ak+1)/(k+2))

as A - + as A - + -

for k = 0, 1, for k

2.

An example of Randol (1966) shows that if there is a zero of order k of the curvature at which the slope of the Jordan curve is rational, then

(k + 1)/(k + 2) may not be replaced by a smaller number, see also Nowak (1984). A classical result of van der Corput (1920) is the following: if p is an upper bound for the radius of curvature of a plane convex body C of class 0°°, then

A(C) - L(C) = 0(p2/3) (12.10) as p ---> + where O( ) may be chosen uniformly for all such convex bodies C. Jarnik (1925) proved that 0(p2/3) may not be replaced by o(p2/3). There are several generalizations of van der Corput's result and Chaix (1972) gave an elementary proof of (12.10). There exist several estimates for the number of points of Z I on the boundary of a strictly convex body C. Andrews (1963) and Chaix (1975) proved that

L'(C) 0. Determine min{). > 0: {)LC + u: u E Zd\{o}, Jul , p} obstructs view from o}.

The orchard problem is the special case where d = 2 and C is the euclidean unit disk. It was solved first by Speiser (cf. Polya (1918)). See also Honsberger (1973) and Allen's (1986) very general treatment of the problem.

Let e = (z, z, ..., z) and let C be a convex body. Cusick (1973) 107

LATTICE POINTS

108

stated the problem to find

min{A > 0: (AC + e + u: u E Zd} obstructs view from o in any

direction r = (r1,

For C = {x: IxJ

.

.

., rd) where r; > 0 for i E {1,

.

.

., d}}.

; for i c- {1, ..., d}} he found that in the cases

d = 2 and 3 this minimum respectively equals and Z. The latter result was found independently by Betke and Wills (1972) in the disguise of a theorem on simultaneous Diophantine approximation. For C the unit

disc Cusick showed this minimum to be equal to 1/U5 and he conjectured that the analogous result for d = 3 was 3/x/21.

Related to the notion of view obstruction is that of `blocking': a family of subsets of E d blocks a point p if p cannot be removed arbitrarily far from its original position along a continuous curve without hitting one of the subsets. L. Fejes T6th (1975(a)) posed the problem of

determining the convex bodies C of minimum volume such that the family of bodies { C + u : u E Z d } does not cover E d and blocks any

point in Ed not contained in the union of the bodies. He conjectured that for d > 2 the extremal bodies are parallelepipeds of volume z and the extremal configurations form a sort of `d-dimensional chess boards'.

For d = 2 L. Fejes T6th (1973, 1975(b)) and Groemer (1966) proved that the extremal bodies are parallelograms and triangles of area z . The solution of the `chess board conjecture' in general dimensions is due to Barany et al. (1986).

A different visibility problem connected with packings of balls was discussed in section 9.6.

13.3 We say that a point v E Zd is visible from u E Zd if V * u and on the open line segment with endpoints u, v there is no point of Zd. A point u E Zd is visible (or primitive) if it is visible from the origin o. A classical folklore theorem says that the proportion of visible points among all points of Zd is where is the Riemann zeta-function.

There exist many counting results of a similar character. We will mention one of them.

For n = 1, 2.... let f(n) denote the minimum number of points of a subset of the set of points u = (u 1, u 2) E Z d with 1 -- U1, U2,---n, such that every point of the set is visible from at least one of the points of the subset. Then a result of Abbott (1974) says that log n 2log log n

f(n) < log n.

Abbott's proof is an existence proof and gives no indication how to

VISIBILITY

109

construct small subsets from which any point of the set is visible. It

would even be of interest to construct such subsets of cardinality O(log n).

Given a finite subset of Z d, can it be translated such that all its points become visible or invisible from o? A satisfying answer to this result is due to Herzog and Stewart (1971): by a pattern P in Zd we mean the following: there exists a positive integer n such that P consists of the points u = (ut, ..., ud) E Zd, where 1 -- ut, ..., ud n and to any

such point there is assigned one of the symbols , +, 0. This gives a representative of P as disjoint union of three sets, P°, P+, P°, say. A pattern P is realizable in Zd if there is a vector v E Zd such that for each point u E P° (resp. U E P+) the point u + v is visible (resp. invisible). Then the following criterion holds: a pattern P in Z d is realizable if and

only if for each prime p we have Zd * P° + pZd (_ {u + pw: U E P°, W E Zd}) (see figure 13.1). 0

0

0

0

0

+

+

+

0

+

0

0

0

+

+

+

0

+

0

0

0

+

+

+

0

+

+

+

0

0

0

0

0

0

0

0

0

0

0

0

(a) non-realizable

(b) realizable

(c) realizable

(v = (1307, 1273))

(v = (53, 19))

Figure 13.1 Patterns

As a corollary we obtain that for d = 2 any pattern consisting only of crosses or of one, two or three squares and any number of crosses can be realized.

13.4 A series of unsolved problems is connected with the `graph of visible

points'. Two points of Z2 are neighbours if one of their

coordinates coincide and the other ones differ by 1. The graph whose nodes are the visible points of Z d and whose edges are the line segments connecting neighbouring visible points is the graph of visible points of Z2 (figure 13.2). By a remark of Herzog and Stewart (1971) this graph is not connected. A close investigation of this graph would be of interest, and there are many natural problems: for example, the number of its components in a

bounded set, paths with particular properties contained in the graph,

LATTICE POINTS

110

11

+

0++ +

+

+

Figure 13.2 The graph of visible points of Z2

density questions, etc. - see, for example, Erdos (1981) or Winfee (1965). See also chapter 16 on lattice graphs.

For more problems of the sort described earlier we refer to Abbott (1974), Erdos (1958), Rearick (1966) and Rumsey (1966).

14

Lattice point problems of integral geometry

14.1 Buffon's needle experiment of 1733 is generally considered as the first landmark of integral geometry. Buffon's result was published only in 1777 (see Buffon (1777)). Despite interesting sporadic results in

the 19th and the beginning 20th centuries, due to Crofton, Czuber, Poincare and others, systematic research in this area started only with the work of Blaschke and his school. To acquaint the reader with the sort of problem considered in integral geometry we state a typical question: what is the `measure' of the set of lines in E d which meet a given convex body? For exhaustive information on integral geometry we refer to Hadwiger (1957) and SantalO (1976). The underlying concepts for integral geometry are topological groups

and Haar measure, but most authors prefer to use the flexible tool of differential forms. In this section we will exhibit several planar lattice point results from

integral geometry by Hadwiger and add some remarks on the higherdimensional case.

In order to formulate two central results of integral geometry due to

Poincare and Blaschke several notions are needed. The kinematic density dK in E2 is the Haar measure on the group of proper rigid motions m = (r(4), t) in E2, suitably normalized. (Here r(t) is a rotation about the origin with angle 0 and t is a translation - see section

10.2.) Equivalently, dK can be defined as the product of Lebesgue measure on [0, 27r] and Lebesgue measure in E2, that is dK = do dt. Let C be a compact set in E2 bounded by finitely many simply closed piecewise smooth curves which can be oriented such that C is on the left-hand side of each of these curves. The Euler characteristic X(C) of C can then be defined as the number of positively minus the number of negatively oriented curves. If C is a polygon, this is equivalent to the definition in section 2.1. If C consists of k disjoint simply connected pieces, then x(C) = k. We shall write A(C) and P(C) for the area and 111

LATTICE POINTS

112

the length of the boundary (perimeter) of C.

Let A, B be two piecewise smooth curves in E2, of finite lengths P(A) and P(B) respectively, and denote the number of points of A n B by n(A n B). Then the formula of Poincare (Poincare (1912), p. 143; Barbier (1860)) says that

f n(A n m(B)) dK(m) = 4P(A)P(B)

(14.1)

where the integral is extended over all motions m of E2 for which A n m(B) * 0. (See Santal6 (1976), p. 111.) Let C, D be two compact subsets of E2, which are equal to the closure of their interiors and with boundaries consisting of finitely many disjoint simply closed piecewise smooth curves. The fundamental kinematic formula of Blaschke (see e.g. Blaschke (1955)) says that

f X(C n m(D)) dK(m) = 27rA(C)X(D) + P(C)P(D) + 2ITX(C)A(D).

(14.2)

Here again the integral is extended over all rigid motions m for which C n m(D) 0 0. (See Santal6 (1976), p. 114.) The results in (14.1) and (14.2) have many applications ranging from the isoperimetric inequality to the following results of Hadwiger (1941).

14.2 Let C, D be compact subsets of E2, each bounded by a simply closed piecewise smooth curve. Let L be a plane lattice and assume that {C - p: p E L} is a packing. (We have written -p only for convenience.) Then, up to boundary points, C is contained in a fundamental domain F of L. The Euler characteristics of C and D are equal to 1. Thus (14.2) yields

27r(A(C) + A(D)) + P(C)P(D) =

f X(C n m(D)) dK(m)

= I L(r,p+t) X(C n m(D)) dK(m) tEF

pEL

fm=(r,t)X((C

- p) n m(D)) dK(m),

rEF

(where X((C - p) n m(D)) = 0 if (C - p) n m(D) = 0). Since C - p and m(D) both are simply connected, their intersection consists of simply connected disjoint pieces. Let k(m) denote the total number of all such pieces of m(D) as p ranges over L. Then

LATTICE POINT PROBLEMS OF INTEGRAL GEOMETRY

22r(A(C) + A(D)) + P(C)P(D) =

JtEF(r, t)

k(m) dK(m).

113

(14.3)

T

Noting that

ftEF

m =(r, t)

dK(m) = 27Td(L),

this formula can be interpreted as follows: the average a of the number

of connected pieces into which a random congruent copy of D is dissected by the packing {C - p: p E L} equals

27r(A(C) + A(D)) + P(C)P(D) 27rd(L)

See figure 14.1. As a consequence of this we obtain that if {C - p: p E L} is a tiling, then there is a congruent copy of D which can be covered by at most [a] tiles. (Note that then A(C) = d(L)). (See figure 14.1.)

Figure 14.1 Integral geometric lattice point problems I

Particular cases of the latter result are the following: A suitable congruent copy of D can be covered by at most 2P(D) A(D) P(D) + A(D) = I1 + 0.636619... [1 + + a2 a2 Ira

1

L

or

squares of edge-lengths a belonging to the standard lattice tiling with squares of edge-lengths a. Similarly, a suitable congruent copy of D can be covered by at most 2P(D) + 2A(D) I1 + V37ra 3\/3a2 11

A(D)] =I1+0.367552... P(D) a +0.384900... a2 L

LATTICE POINTS

114

regular hexagons of edge-length a which belong to the standard lattice tiling with regular hexagons of edge-length a. By covering the hexagons with circular discs, one obtains a similar covering result with discs. There exist higher-dimensional analogues of these results of Hadwi-

ger, due to Santald (1944), but the proofs seem to hold for convex bodies only - see Groemer (1986). In order to give the reader a feeling for these extensions we cite the following result: any convex body C in E d can be covered by at most 1 +

V(C) ad

1

d-1

+ - E Kd

1

x; i

Wd-i(C)1 J

a'

cubes of edge-lengths a, all of which belong to a suitable lattice tiling. Here Ki denotes the volume of the solid eculidean unit ball in E' and Wi(C) is the ith quermassintegral of C - see section 3.6. It remains an open problem to extend this to topological balls.

The results of this subsection are of importance for results of Groemer (1986) on multiple packings - see section 8.7.

14.3 Let A, B be two piecewise smooth curves in E2 of finite lengths. Assume that L is a lattice and that A is contained in a fundamental domain F of L, where by a fundamental domain of L we understand a subset F of E2 such that for each x E E2 there is exactly one p E L such

that x + p E F; note that this is slightly more precise than the corresponding concepts of fundamental parallelotope, fundamental domain and simple reduction domain defined respectively in sections 3.1, 10.4 and 11.2. For a rigid motion m let 1(m) denote the number of points of the intersection of B with the (disjoint) curves {A + p: p E L}. Then reasoning similar to that which led to (14.3) but using (14.1) instead of (14.2) shows that

4P(A)P(B) =

J m-(r,t) teF

l(A n m(B)) dK(m).

This permits the following interpretation: if a random congruent copy of B is put on the plane, the mean value of the number of points which this curve has in common with the system of curves {A + p: p E L} is equal to 2P(A)P(B) = P(A)P(B) 0.636619. 7rd(L) d(L) See figure 14.2. If, for example L has an orthogonal basis {b(1), b(2) } with I b (1) J = a, l b (2) = i and A is the polygonal line with vertices b (1) , o, b (2), then the mean value of the number of points which an arbitrarily

LATTICE POINT PROBLEMS OF INTEGRAL GEOMETRY

115

placed congruent copy of B has in common with the double grid of lines corresponding to the basis {b(1), b(2)} of L is

2(a + r)P(B)

T

= 0.636619.. a + P(B). (14.4) Iroi ai By taking for B a line segment of length A i and letting a - +oo we see that the probability that a randomly chosen line segment of length A meets any of a system of parallel lines which are at distance A apart is 2A

=0.636619.....

This is the classical result of Buffon.

The mean value in (14.4) can be used to determine the approximate length of a curve B by placing at random a double grid of orthogonal lines on B and counting the number of intersection points. See figure 14.2.

Figure 14.2 Integral geometric lattice point problems II

15

Applications to numerical analysis

15.1 The notion of lattice and several lattice point results are important for purposes of numerical analysis and applied mathematics. In this section we first introduce zeta-functions on lattices and discuss their relevance for numerical integration. We then present the basic idea of the multi-grid method for boundary-value problems of partial differential equation. We have inserted the multi-grid method in this book for two reasons. The first is its recent importance in numerical analysis.

The second is that the multi-grid method again exhibits the fact that even very simple ideas about lattice points may have far-reaching consequences in other areas. (A different example of this is Minkowski's

fundamental theorem which applied to many problems of number theory.) We are optimistic that there exist many more applications of lattice point results in other branches of mathematics.

15.2 A direct generalization of the Riemann zeta-function is the following concept of zeta-function for lattices, first introduced in a different context by Epstein at the beginning of this century and more recently defined independently by Sobolev: IEL\{o)

1

ill's

for lattices in L in E' and s > d/2. Here I I denotes the euclidean norm on Ed.

The lattices of given determinant for which, for a fixed value of s > d/2, the zeta-function attains its minimum have been determined so far for d = 2, 3 only. For d = 2 this was done by Rankin (1953), Cassels (1959), Ennola (1964(a)) and Diananda (1964). The extremum lattice is unique up to rotations and has a basis (b('), b(2)) such that b (1) I = I b,2)1 = I b (') - b (2) 1, i.e. it is an `equitriangular' lattice. In case 116

APPLICATIONS TO NUMERICAL ANALYSIS

117

d = 3 Ennola (1964(b)) proved that the extremum lattice is unique up to rotations and gives a lattice packing of balls of maximum density. For several further properties of the zeta-function the reader may consult the article of Delone and Ryskov (1967). For large values of s only the points of L\{o} which are closest to o contribute significantly to (L, s). Hence if L provides a local minimum

of (L, s) among all lattices of given determinant and arbitrarily large values of s, the points of L\{o} closest to o must be as far away from o as possible. This shows that L provides a locally densest lattice packing of euclidean balls.

15.3

The importance of the zeta-function for numerical integration seems to have been first recognized by Sobolev, see Sobolev (1974): Consider the space of all real functions f on E' with compact support contained in a fixed bounded domain D and with bounded continuous partial derivatives up to order s. How should one choose nodes x(l), . . ., x(k) E D such that the error which one makes when replacing the integral fdxl...dxd JD

by an expression of the form V(D) y, Ax (0) k ,_,

is - in a well-defined sense - as small as possible? For d > 2 it seems to be hopeless to give an answer, unless some additional assumption about

the nodes is made. If the nodes are the points of a lattice L of given determinant, say S, which are contained in D, then, neglecting the contribution of the points near the boundary of D, the best choice for L is when the following is obtained: for the given value of s the c-function attains its minimum on the set of all lattices of determinant 1/6 for the polar lattice L* of L. The results described before thus show that for

d = 2, 3 the `best' lattices are the `equitriangular' lattices and the lattices which provide the thinnest covering of E3 with balls, i.e. the `body centred' cubic lattices.

15.4 For different classes of continuous functions the problem of numerical integration was considered by Babenko (1976, 1977) using Dirichlet-Voronoi cells as a tool. We give a description of a special case

of one of Babenko's (1976) results: Let D be a bounded Jordan

LATTICE POINTS

118

measurable set in Ed. For 0 < p < 1 consider the class 'P of real functions f on D which are Holder continuous in the following sense:

for x,yED.

If(x) - f(Y)I , (I Ix - YIIm)P

Here I . denotes the maximum norm: I Ix I x E Ed. Then I

I

I

1

s

f dxl ... dxd - V(D)

UPp

k

If(x( =1

I

= max{ Ix 1 1, dP (D)

... ,

1+pld

2 (d + p)

Ixd I } for P1

k

Id

for any choice of x(l), ..., x(k) E D. If the nodes x(1), ..., x(k) are (essentially) the points of

(vD))lIdzd nD then the lower bound on the right-hand side is asymptotically best

possible as k -. A simple refinement and generalization of Babenko's result is due to Gruber (199?).

15.5 There exist many discretization methods in numerical analysis by means of which various types of (partial) differential equations and corresponding boundary-value problems are transformed into systems of linear equations for the (approximate) values which the solution of the differential equation assumes at certain points of some fixed lattice (see e.g. Collatz (1966)) and Gerald and Wheatly (1985). Now the problem arises of finding the solution of the system of linear equations. Consider an approximate solution. The error, that is the difference between the exact and the approximate solution of the system of linear equations, is

decomposed into a 'high-frequency' and a `smooth' component. To reduce the error of the high-frequency component the given grid is used whereas for the smooth component a coarser subgrid is sufficient. This reduces the number of steps required to find an approximate solution of the linear system of equations having a smaller error (two-grid method).

Iterating this idea leads to the multi-grid method. (See Hackbusch (1985) and Jespersen (1985).) The multi-grid method was first introduced by Fedorenko (1964) in the case of Poisson's equation. Since for

more than one variable the presentation of the multi-grid method becomes technically involved we will consider a one-dimensional model

case first and then indicate the necessary modifications in the twodimensional case. Our exposition follows Hackbusch (1985). Consider the 1-dimensional Dirichlet.boundary-value problem

-u"(x) = f(x)

on (0, 1) and u(0) = u(1) = 0,

(15.1)

APPLICATIONS TO NUMERICAL ANALYSIS

119

where f is a given real continuous function on (0, 1). For the level 1 = 0, 1/21+1. The grid of level I consists of the points 1, 2, ..., let h, = xi = ih,, i E (1, 2, ., 21+1 - 1). .

.

Next, at every grid point xi replace the derivative in (15.1) by a suitable difference expression, for example by

h (-u(xi - h1) + 2u(xi) - u(xi + h,)) = -u"(xi) + 0(h 2). Setting

ur = (u(xl), ..., u(x2,.'_l))", fl = (f(x1), ..., and omitting the O(hl)-terms, this gives the following linear system for u1:

Lrur = fl,

(15.2)

where

2 -1 2 -1

-1 I1 =

2

-1

h1 l

2 -1I

-1

2

The matrix L1 of the system (15.2) is tridiagonal and thus (15.2) may be solved easily in a direct way. Since our aim is to show the basic idea of the multi-grid method we proceed instead as follows. For the solution of (15.2) one can use for example the Jacobi iteration starting with some vector us°): 2

u(k+1) = 11

hr (L1ulk) 2

- fl),

or the damped Jacobi iteration with damping factor 2, say, ur

k+1) = usk)

- hr 4

'k) -

(Lrur

4 Lr) ur hr

(k)

h

+ 4 fr

(15.3)

Here I, denotes the (21+1 - 1) x (21+1 - 1) unit matrix. The eigenvalues and the eigenvectors of the iteration matrix I, - (h2/4)L1 are A11 = 1 - sin2(jTrh1/2),

e(,i) = Ugh, (sin(jnhr),

..., sin((21+1 - 1)Jn.hr))", jE{1,...,21+1-1).

Since the maximum of the absolute values of the eigenvalues, the so-called spectral radius, 1 - 7r2h1/4 + 0(h°), is almost equal to 1, the overall convergence of the damped Jacobi iteration is slow.

LATTICE POINTS

120

If one represents the initial error u}") - u, of the damped Jacobi iteration in the form

u,°1 - u, _

aieli)

then k steps of the damped Jacobi iteration yield the error l ail for j small, u, _ /3iey), where lf3i l = l ai l IA,i I k {l 1 be given and let G be the lattice graph the vertices of which are precisely the points v 1 E Z d with 1 < v 1: n 1 for i E {1, . . ., d} and where two vertices are joined by an edge if their

(euclidean) distance equals 1. (One may think of an edge as a line segment of length 1.) A result of Kotzig (1964) states that G has a hamiltonian circuit if and only if n1 nd is an even number. Several more properties and applications have been found on hamilto-

nian circuits for lattice graphs by Kurschak (1927), Nash-Williams (1960), Kotzig (1964), Thompson (1977), Binz (1978), Myers (1981),

Cannon and Dolan (1986) and many others are scattered in the literature on recreational mathematics, disguised as puzzles, board games, etc. One such problem is the `knight tour problem' on a chessboard, using the legal moves only, visiting every square exactly once. How many different tours can be designed?

16.3 Many problems of chess can be translated into lattice graph problems and can sometimes be solved by graph-theoretic machinery. An example is the problem how to place the maximum number of queens on the chessboard so that no two attack each other. (It is interesting to note that this problem was known already to Gauss.) Consider any chess piece P (not a pawn) and a lattice graph G whose

vertices correspond in an obvious way to the 64 (or in general, n) squares of a (possibly rectangular or even more general) chessboard. Two vertices u, v are joined by an edge if and only if P placed on the square corresponding to u can attack the square corresponding to v. The problem now is to find a set S of vertices of G with the properties that 1.

no two vertices in S are adjacent and

LATTICE GRAPHS

125

the number of vertices in S is maximal.

2.

For solutions to this problem and relevant references see Foulds and Johnston (1984).

Originally most of these problems were designed as puzzles and games. Later people found interesting applications in many parts of combinatorics. Perhaps the best known of these applications is the rook polynomial.

For a given chessboard with n squares let rk, k = 0, 1, 2, ..., denote the number of ways of placing k rooks on the board in such a way that no two attack each other. Define its rook polynomial r by n

r(x) _ I rkxk

for x E R.

k=0

The rook polynomial and some of its extensions are related to several parts of combinatorics, for example, the principle of inclusion-exclusion, permutations with forbidden positions, latin rectangles and matching problems. For particulars see Kaplansky (1946), Riordan (1958) and Roberts (1986).

16.4 Let S be a finite set of points in the plane. A tree in the plane whose vertices include all points of S and whose edges are non-crossing

line segments is a Steiner tree (figure 16.1) on S if the sum of the (euclidean) lengths of all edges is minimal (among all such trees). Let L(S) denote this sum. Steiner trees have a long and venerable history, dating back at least to Maxwell. They have a wide range of applications in optimization. For information see Gilbert and Pollak (1968) and Maculan (1987).

Figure 16.1 Steiner tree

Let

L(n) = max L(S) taken over all sets S of n points in the unit square.

L. Fejes TOth (1940) showed, by considering points arranged in a

LATTICE POINTS

126

regular hexagonal lattice, that

L(n) ? (3/4)1/4 \/n + 0(1). On the other hand Chung and Graham (1981) obtained for sufficiently large n

L(n) < 0.995Vn. They conjectured that

L(n) < 0.99\/n. They also considered the corresponding problem when the edges of the tree are parallel to the sides of the square and obtained

Vn + O(1) < L*(n) < 1/n + 1 + o(1) and conjectured

L*(n) !S Vn + 1

for all n. (L*(n) is defined similarly to L(n) but using the linear rectilinear distance Ix-y111=Ix1-x21+Iy1-Y21.)

Even the latter conjecture seems so complicated that it has not been verified even in the case when the points are the points of tZ2 contained in the unit square. Gardner (1986) exhibits trees on checkerboards of several different sizes but is unable to prove that they are Steiner.

16.5 Several important quantities in combinatorics can be interpreted as the number of certain lattice paths. These interpretations lead in many cases to very simple proofs of combinatorial identities. The following interpretation of binomial coefficients was apparently

first introduced by P61ya (1962). By a (staircaselike) path in Z2 we understand a finite sequence of vertices u(°) = o, v(1), U(2), ... such that

fork=0,1,2,... V(k+l)

= (Ulk) + 1, v?)), or (v

,

Uzk) + 1),

where U(k) = lUlk), V?));

then the number of paths from o to (x, y) e Z2\{o}, x, y ? 0 is xxy).

(If this number is denoted b(x, y), then b and the binomial coefficients satisfy

LATTICE GRAPHS

1y)+(x+X

0/-1'11y(XXI=(xX

b(0,y)=1, b(1,y)=y,

127

b (x, y) = b (x - 1, y) + b (x, y - 1)

for (x, y) E Z2, x, y ? 1 and hence coincide.) Since the number of path

from o to (x, y) equals the number of paths from o to (y, x) the identity (XY+ Y

xxy) is obtained. With similar ease one can derive many other identities involving binomial coefficients using this method. Next we consider the more general concept of a (non-decreasing) path

in Z2, viz. a finite sequence of points 0) = o, U(1), v(2), ... such that for k = 0, 1, 2, .. U(k+1) = (U(k) + 1, V2(k) + 1) or (V (k), v + 1) or (v + 1, v.k) + 1) .

for v(k) = (Ujk), V M.

Kasinsky and Bryant (1983) denote the number of these paths from o

to (x, y)EZ2\{o}, x, y?0by

\xxy) and derive a number of identities for these numbers, for example

x- 1

x

x- 1

x

for x, y ? 1, (x - 1, y - 1) E Z2\{o}. Thus one can arrange the values of

x in

X

yl

a triangular array similar to Pascal's triangle for the binomial

coefficients but with a different rule for the passage from two consecutive rows to the next one: 1

3

1

/

1

\\

5

5.

1

I:

1

UV

\

1

7

13

7

1

LATTICE POINTS

128

The numbers and

\x X y) are related. By considering the paths with 0, 1, 2, ... `diagonal edges' one arrives at the relation

E (x zy k2k)(x+y-k). k=O

Similarly one can show that ((1x+y-k).

( xxy)-E1k) Related but different arrays of `triangular number' were introduced by Shapiro (1976) and D. G. Rogers (1978).

Numbers based on paths the edges of which join mutually visible lattice points were introduced by Mohanty and Handa (1968). They are related to multinomial coefficients. More material of this sort including extensions to higher dimensions and lattice-path interpretations of other

problems can be found in Sved (1984), Breach (1985), Garsia and Remmel (1985) and in the monographs of Mohanty (1979), Narayana (1979) and Goulden and Jackson (1983).

16.6 It is well known how much the so-called `spreadsheet' is applied in computing. A spreadsheet is a two-dimensional square lattice which appears on the screen of a cathode-ray tube of a computer. The squares are usually called cells. The `value' of each cell can be made to depend on any other cell or a group of cells. In general the cells can be used to describe mathematical expressions or some logical relations. The spreadsheet provides a suitable medium for handling diverse problems from accounting to mathematical games. In this section we discuss only the relationship between a continuous region and its computer image. As an example we define a digital convex set and state a digital version of Helly's theorem. For other results we refer to the following sources: Minsky and Papert (1968), Sklansky (1970), Doignon (1973), Rosenfeld and Kak (1976), (1982), Sklansky and Kibler (1976), Kim (1981), Kim and Sklansky (1982), Hammer (1984), Arganbright (1985) and Horn (1986).

A digital region is a finite set S of integer lattice points. A finite set of lattice squares in E2, i.e. of squares of edge length 1 whose vertices

belong to Z2, is called a cellular region. It is clear that there is a one-to-one correspondence between digital and cellular regions by

LATTICE GRAPHS

129

assigning to each point u E Z2 the lattice square with u at its lower left vertex.

Given a `continuous' object C in E2, such as a compact set or a convex body, its digital (resp. cellular) image is the digital region

CnZ' (resp. the set of lattice squares which have non-empty intersection with C). Instead of C a computer `sees' the digital or cellular image of C. It thus seems to be important to define geometric and topological properties of digital or cellular regions in terms of their `continuous preimages'.

If a, b are two points of a lattice, then the digital line segment with endpoints a, b is the intersection of the lattice with the closed continuous line segment [a, b]. A digital region S is digitally convex if there exists at least one convex

set of which the digital region is the digital image. In other words, S is digitally convex if it is the intersection of the lattice with a convex set. This definition implies that for a digitally convex set S the digital line segment with endpoints a, b is contained in S whenever a, b E S. (The converse of this statement does not hold, as may be seen by considering the digital region S = {(0, 0), (1, 2), (2, 1)).) Doignon (1973) proved the following remarkable digital version of

Helly's theorem in dimension d: A finite collection of at least 2d digitally convex sets has a non-empty intersection if each subcollection of 2d members has a non-empty intersection. (The number 2d is best possible.)

References on digital processing are Horn (1986) and Rosenfeld and Kak (1976, 1982).

16.7 In this section we present problems which have interpretations in physics and other sciences. For many more such problems see the surveys and monographs cited below. Some of these problems can be interpreted in terms of covering, packing or tiling. We use these interpretations, for they prevail in the literature.

1. The cell growth problem. A one-celled animal is simply a lattice square. A one-celled animal can `grow' by successively adding further one-celled animals to its free sides. In this way one obtains n-celled animals for n = 1, 2, . ., also called polyominoes. How many simply (or multiply) connected n-celled animals (for given n) are there up to isometries of the plane? For problems of this sort and generalizations to animals consisting of regular triangles or hexagons we .

LATTICE POINTS

130

refer to Golomb (1965, 1966), Harary (1967) and Temperley (1981).

An appealing tiling result on four-celled animals is due to Coppersmith (1985). It says that one can tile the plane with congruent copies of any four-celled animal. The corresponding problem for five-celled animals remains open, while there are six-celled animals which admit no tiling of the plane. 2. The dimer problem. If a biatomic gas molecule of the right size is adsorbed on the surface of a crystal it occupies two neighbouring sites.

Thermodynamical problems of systems of such molecules led to the following mathematical problem: Consider a lattice graph G in Z2 having n vertices and such that two vertices are adjacent precisely if they have distance 1. A dimer on G is a subgraph consisting of two adjacent vertices and the edge joining them. A dimer covering of G is a spanning subgraph of G each vertex of which has valency 1 (this means that it consists of disjoint dimers which contain all vertices of G). The problem is to determine the number of different dimer coverings of G or to give asymptotic estimates for increasing sequences of Gs. One can show that if m, n are non-negative integers then (using some difficult analysis) for the rectangle {u E Z2: 1

(17.3)

fan/(logn)1/2.

(In this section all a, /3, y, 6, a, i are positive constants.) It was shown that if (17.2) and (17.3) are true, they are best possible. Also it was proved that

(n - 1)1/2 - 1 < f(n) < an/(logn)]n.

(17.4)

The upper bound in (17.4) is obtained by considering the number of lattice points {(x, y): 0 < x, y < Vn} (see Erdos (1946)). L. Moser (1952) improved the lower bound in (17.4) to 6n2/3 and Chung (1984) improved this to bn5/7. For g(n) it was proved that n1+r/loglogn < g(n) < 2n2/3.

Again the latter is obtained by considering the lattice points {(x, y): 0:5 x, y "Vn). See Erdos (1946). Beck and Spencer (1984) improved the first inequality to

3. Let {x('),

...,

n312-,, for a certain E > 0.

x(n)} be a set which implements f(n), i.e. D(x(1), x(n)) = f(n). Is it true that {x(1), ..., x(')) has lattice structure?

The first step to answer this question would be to decide if there is always a line which contains an 1/2 of the x Ws (and in fact n' instead of

n1/2 would be interesting), where a > 0 is an absolute constant. A stronger result would be that there are an 1/2 (or n1-E) lines which contain all the x0s. The only result in this direction, due to Szemerddi,

states that if D(x(1), ..., x(n)) is o(n) (more precisely we should consider a sequence of sets {x(n'1) x(n'n)}, n = 1, 2, ..., but we adopt the usual sloppy notation) and k is a positive integer, then there

is always a line which contains at least k of the points supposing n is sufficiently large (see Erdos (1975)). In fact Szemerddi's result gives that

such a line can be chosen as the perpendicular bisector of two of our points, and also that there are o(n) lines which contain all our points. 1

4. Erdos and Guy (1970) ask: how many lattice points (u;'), up), k with coordinates 0 < u;'), u(j) n may be chosen with all

mutual distances distinct? They conjecture k < an 2/3 (log n) 1/6

and showed that n2/3-E < k < an /(log n) 1/4 for any E > 0 and sufficiently large n.

EXTREMAL COMBINATORIAL PROBLEMS

135

Further related problems can be found in Erdos (1975, 1986) and in Moser and Pach (1986).

17.2 A well-known theorem of van der Waerden (1927, 1971) states that for all positive integers 1, h there is a number w(l, h) such that if the positive integers not exceeding w(l, h) are partitioned into l classes, at least one class contains an arithmetic progression containing h terms.

The geometric interpretation of the theorem makes it plausible to extend this to higher dimensions: For any finite subset H of the lattice points of Ed and any partition of the lattice points of Ed into 1 classes,

at least one class contains a subset which is homothetic to H. This theorem was established by Grunwald (see Rado (1938)). There are several other generalizations and extensions of van der Waerden's theorem - see Erdos and Graham (1980), Guy (1981) and Sos (1983) and the references there. For the purpose of obtaining estimates for the number w(l, h), Erdos and Turan (1936) introduced the quantity rk(n), defined to be the least

integer r so that if

l 0. This is a discrepancy-type problem in the theory of uniform distribution (see Sos (1983)). A basic problem in the theory of uniform distribution is to colour the elements of a set with two colours as uniformly as possible with respect to a given family of subsets such that each colour meets each subset in approximately the same number of elements. Now, let X be a finite set and X be a family of subsets of X. Given a two-colouring f: X -> {+1, -1 } of X, and a set SEX , call

E = E(S, f) = Zf(y)l yES

the error off relative to S. Set

D(X), f) = max(E(S, f)); S E X} and define the discrepancy Disc(X, f) of f by

Disc(X, f) = min(D(X), f); f a two colouring of X}. The basic problem is to estimate and, if possible, to determine Disc(X)). Beck (1982) conjectures that

Disc(i) = O((a log a)1/2) where

a = max IS E ti: y E SI. YEX

Beck (1982) has the following generalizations of Roth's theorem: If we

LATTICE POINTS

138

{(ul, u2): the lattice points of the n x n-square 0 < u; < n} C Z' with red and blue in any way (n = 1, 2, 3, ...), there exists a line segment with E at least /3n'/'-f, where (3 > 0 is a suitable absolute constant. On the other hand there exist two-colourings of the n x n-square (n = 1, 2, 3, ...) such that for any line segment colour

E = O(n'13(log n)'I'). A more general theorem of Beck (1985) is the following: There is an absolute constant y > 0 such that for convex bodies C in E' for which the radius of the largest inscribed eucidean ball is at least 1 and for any colouring f: Zd -b {1, -1} there is a proper orthogonal transformation A, a real number A E [0, 1] and a vector t E E d such that

E flu): u c- (AA(C) + t) n Zdl ? yS(C)'/'. Here S(C) denotes the surface area of C. Uniform distribution and, in particular, results dealing with discrepancy have applications in geometry, number theory, numerical analysis and several parts of combinatorics. See Sos (1983), Beck (1985) and Lovasz et al. (1986). For general information on uniform distribution we refer to the books of Kuipers and Niederreiter (1974) and Hlawka (1984).

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