Israel Journal of Mathematics, Volume 3, Issue 1 (1965)

ON THE A P P R O X I M A T I O N OF S Y M M E T R I C OPERATORS BY OPERATORS OF F I N I T E R A N K BY

SHMUEL KANIEL* ABSTRACT

The following Theorem 1 and Theorem 2 are established. The proof utilizes the elementary properties of symmetric operators. It is shown that the symmetry condition is necessary for these theorems to hold. In this work we shall prove the following theorem which stems from consideration of the minimal iteration method for eigenvalue problems ([1], [2]). TrmOREM 1. Let A be a bounded symmetric operator in Hilbert space H. Let f ~ H f ~ 0 be chosen and let H k denote the span of f, Af,...,Akf. Let e be the orthogonal projection on H k and let B denote the restriction of P A to H k. Then there exists an absolute constant c(k) tending to zero as k ~ co which does not depend on A nor on f so that for some 2 ~ a(A) and some # ~ a(B): 2

[2 - "1--< c(k)I1 All-~ 4(~ 1---7 + IIa II"

(1)

The proof of Theorem 1 depends on the following propositions:

PROPOSITION 1.

if

IIAT-,fll---~llfll,

then there exists 2 ~ a ( A ) s o

that

Ix-.l__~-.

SOME REMARKS ON NUMBER THEORY

7

(4) immediately implies A k ~_ n/4, thus by k < 10 n (2) follows, and our Theorem is proved. Perhaps Ak > ck holds for infinitely many values of k*. In this connection I would like to mention the following question: Denote by f ( n , c ) the smallest integer so that if I z~l > 1, 1 < i < n are any n complex numbers, there always is an integer 1 < k < f(n, c) for which

I

t=!

>__c

A very special case of the deep results of T u r i n [8] is that f ( n , 1) = n. R~nyi and I [3] obtain some crude upper bounds for f ( n , c,) if c > 1, but our results are too weak to improve (2). II. Is it true that to every ~ > 0 there is a k so that for n > no every interval (n, n(1 + ~)) contains a power of a prime Pi < Pk? It easily follows from the theorem of Dirichlet quoted in I that the answer is negative for every ~ < 1, since the above theorem implies that to every ~ > 0 there are infinitely many values of m so that all primes Pi --< P~ have a power in the interval (m, m(1 + t/)) and then the interval (m(1 + t/), 2m) must be free of these powers. Let us call an increasing function g(n) good if to every t / > 0 there are infinitely many values of n so that all the primes p~ < g(n) have a power in (n, n(1 + t/)). It easily follows from the theorem of Dirichlet and ~ ( x ) < c x / l o g x that if (5)

[loglog n • logloglog n) g(n) = o ~ ~ g ~ g l o - - ~ n

then g(n) is good. I leave the straightforward proof to the reader. I can obtain no non-trivial upper bound for g(n). Letl 1. (5) and ~ p < y l / p = l o g l o g y + 0 ( 1 ) implies that for infinitely many n (7) A(n, a) > loglogloglog n + 0(1). Now we are going to prove (8)

lim inf A(n, ct) = O. B=t:O

To prove (8) we shall show that to every 8 > 0 there are arbitrarily large values of n for which (9) A(n, o~) < 5. • By a remark of Clunie, we certainly must have c __< 1. Added in proof: Clunie proved f(n,c) < g(e) n log n, A k > c k ½

8

P. ERDOS

[March

Let k = k(e) be sufficiently large. Consider ~'A(21,~) where in ~ ' the summation is extended over those l, 1 < ! < x for which the interval (2 t, ~2~) does not contain any powers of the primes Pi, 1 < i _< k. Put D(~, k) =

1

log Pk

i = 2

.

Let ax, "", ak be positive numbers which are such that for every choice of the rational numbers rl, ..., r k not all 0, ]~=lria~ is irrational. The classical theorem of Kronecker-Weyl states that if we denote by x,, 1 < n < ~ the point in the k dimensional unit cube whose coordinates are the fractional parts of n~i, 1 _< i _< k then the sequence x, is uniformly distributed in the k dimensional unit cube. From this theorem is easily follows that the number of summands in ]~'A(2', a) is (1 + o(1))xD(oq k). Thus to prove (9) it will suffice to show that for every suffÉciently large x £

]~ 'A(f, ~z) < -~- D(~, k)x.

(lO)

We evidently have ]~ u(j, x) pk k / 2.

The Lemma can be deduced from 1.6] without any difficulty. Now we can prove our Theorem. It is easy to see that if n does not have property P then it is included in a unique maximal interval of consecutive integers no one of which is relatively prime to the others. Denote these intervals of consecutive integers by I1, I2 "" where/1 are the integers 2184, 2185... 2200. Let I, be the last such interval which contains integers < x. l I I denotes the length of the interval I. To prove our Theorem it suffices to show

(18)

i=l

IIJl <X(1--C2)

Clearly none of the intervals Ij contain any primes. To prove (18) it will suffice to show that for some ca < cI (19)

•311j[ < (ci - c3)x

where cl is the constant occuring in Lemma 1 and in ~3 the summation is extended over those I j, 1 ~ j < r which are in the intervals (pj, p j+ 1) satisfying (17). Let T be sufficiently large and consider in the intervals (17) those integers all whose prime factors are at least T. It easily follows from Lemma 1 and the Sieve of Eratorthenes that the number of these integers not exceeding x is at least (20)

(1 + o(1))clx I I

(1 - 1 / p ) > c 4 x / l o g T

p

r.

SOME REMARKS ON NUMBER THEORY

19651

11

Z < Iljl < (1 - e) log x.

(22)

Write

~,,lljl

(23)

= IE

~g~ltjl

r

where in ~ ' ) we have (r = 0,1...)

2"r< Itl__< 2'+1T

(24)

if2"+lT>(1-e)logx, then the upper bound in (24)should be replaced by (1 - e)log x. Now we show that for sufficiently large T and every r (25)

V(,)I ,-,4 I I J 1I < 2x / (2'T) ½.

From (25) and (23) (21) easily follows for sufficiently large T. Thus to prove our Theorem we only have to show (25). The integers in the Ij of ~ ' ) can not be relatively prime to N2,+I.T (Nk is the product of the primes not exceeding k) therefore if Ij is in an interval (uN2,+,. T, (U + 1)N2r+l.r ) Ij must lie in an interval (ai + uN2,+~.r, ai+ ~ + uNz,÷~.T) where 1 = a 1 < ... < %(Nz,+l.T ) =

N2,.+I.T --

1

are the integers relatively prime to N2,.+ ~.T" Since 2 "+ 1T =< (1 -- e) log x, it follows from the prime number theorem that N 2 . . . . r = O(X), hence we easily obtain from Lemma 2 for sufficiently large T ,,,


log x and thus can not prove that the density of the integers having property P exists. COROLLARY. There are infinitely m a n y composite integers satisfying (16). By % > 0 there are infinitely many composite integers having property P, and if there would be only a finite number of integers with property (1) then for sufficiently large i in the set of integers p~ < t < Pt+i no one would be relatively prime to the other, thus only a finite number of composite integers would have property P. This contradiction proves the corollary. Let us say that the primes have property P0, the composite integers satisfying (16) have property P~. By induction with respect to k we define: An integer n has property Pk if it does not have property Pj for any j < k, but both intervals (n, n + u(n)) and (n - u(n), n) contains an integer having one of the properties

12

P. ERDOS

P j, 0 < j < k. It is easy to see that for every k > 0 the integers having property Pt have property P too, and conversely every integer having property P has property Pk for some k > 0. It is easy to show by induction with respect to k that the integers having property Pk have density 0, hence from ~p > 0 we obtain that for every k there are infinitely many integers having property Pk. REFERENCES 1. A. Brauer, On a property of k consecutive integers, Bull. Amer. Math. Soc. 47 (1941), 328-331: Sivasankaranarayana Pillai, On m consecutive integers III, Prec. Indian Acad. Sci. Sect. A, 12 (1940), 6-12. 2. P. Erd6s, Problems and results on diophantine approximation, Compositio Math., 16 (1964), 52-65, see p. 52-53. 3. P. ErdiSs and A. R6nyi, A probabilistic approach to problems of diophamine approximation, Illinois J. Math., I (1957), 303-315, see p. 314. 4. P. Erd/Ss, The difference of consecutive primes, Duke Math J., 6 (1940), 438-441. 5. P. Erd6s, Amer. Math. Monthly, 60 (1953), 423. 6. G. Hoheisel, Primzahlprobleme in der Analysis, Sitzungsber., Berlin (1930), 580-588. 7. C. Hooley, On the difference of consecutive numbers prime to n, Acta Arith., 8 (1962-63), 343-347. 8. P. Turfin, Eine neue Methode in der Analysis und deren Anwendungen, Budapest 1953. TECHNION-ISRAELINSTITUTEOFTECHNOLOGY, HAIFA

COMPACTNESS AND SEMI-CONTINUITY BY

S. P. F R A N K L I N ABSTRACT The purpose of this note is to point out that a recent result of Ceder yields easily converses t o well known theorems of Wallace and Birkhoff and thus provides two new characterazations of compactness as well as sl~ecifying the class of spaces for which the theorems are true. A very slight extension of Ceder's theorem is also obtained, as well as new and simple proofs.

Let X be a non-empty topological space and P a partially ordered set. A function f : X ~ P is an upper-(lower-)semicontinuous function iff x~ ~ x 0 and eventually f ( x , ) > c( < c) implies that f(xo) > c ( < c). The theorem of Birkhoff ([1], p. 63) asserts that when X is compact, (B) every upper-(lower-)semicontinuous function to a partially ordered set assumes a maximal (minimal) value. A quasi order < (reflexive and transitive)on a non-empty topological space X is upper-(lower-) semicontinuous ifffor each x ~ X, {y E X lx < y} ({y e x ly - b} - cover X. For each x ~ X, let f ( x ) = {b [x ~ l/b} ~ D. Suppose in X a net {x,} converges to xo and eventually f ( x , ) ~_ C c D. If a ef(xo) I C, eventually x~ e Fo contradicting f ( x , ) ~_C. Hencefis a lower-semicontinuous function into ~0(D), partially ordered by set inclusion, which assumes no minimum. REFERENCES

1. G. Birkhoff, Lattice Theory, Amer. Math. Soc. Colloquium Publications, 25, rev. ed. (1948). 2. J. G. Ceder, Compactnessandsemicontinuous carriers, Proc. Amer. Math. Sot., 14 (1963), 991-3. 3. E. Michael, Topologieson spaces of subsets, Trans. Amer. Math. Sot., 71 (1951), 152-82. 4. A. D. Wallace, A fixed point theorem, Bull. Amer. Math. Soc., 51 (1945), 413-16. 5. L. E. Ward, Jr., Partially ordered topologicalspaces, Proc. Amer. Math. Soc., 5 (1954), 144--61. UNIVERSITY OF FLORIDA AND CARNEGIE INSTITUTEOF TECHNOLOGY

UNION CURVATURE OF A VECTOR FIELD IN A SUBSPACE* BY

R. N. KAUL

ABSTRACT

The differential equations of union curves on a hypersurface V~ immersed in a Riemannian V,+I have been obtained by Springer [1]. These results were generalized later for a subspace in a Riemannian space by Mishra [2]. The author [3] has defined the union curvature of a vector field with respect to a curve on a hypersurface If. of a Riemannian V~+t. The purpose of this paper is to consider union curvature of a vector field with respect to a curvein a subspace V. of a Riemannian V,..

1. Subspaces of

Vm .

Totally indicatrix variety of a vector field.

Consider a subspace Vn of coordinates x~(i = 1 , . . . , n ) (1.1)

and metric

ds 2 = g~jdxtdx J

immersed in a Riemannian V, of coordinates y ~ ( a = l , . . . , m ) and metric (1.2)

ds 2 = a~t~dy~dy p

Let N~/(tr = n + 1,..., m) be the contravariant components in the y ' s of (m - n) independent mutually orthogonal unit vectors normal to V~, then the following relations hold [4] (1.3)

~ p = 1 a~pN~/N~!

(1.4)

• P = 0 ao,aN~/N~!

(It # a)

(1.5)

a~pN~/ya, i = 0

(tr = n + 1, ..., m)

(1.6)

y;~j = •

I'~,/,IN~*/

where the coefficients fl,/~ are symmetric covariant tensor of the second order. Let Vbe a field o f unit vectors in Vn such that if x t = xt(s) defines a curve C in V~, there is associated a unit vector with each point of the curve. I f v~ and p~are the components of Vin the y ' s and the x's respectively, then v~= y ~,~p. • The author is indebted to the referee for helpful suggestions. Received March 5, 1965. 15

16

R.N. KAUL

[March

We have the following relation for the derived vector T ~ o f Fin Vmwith respect to the curve C [3,4,5]

/

i dxj\

•

I o~

T ~ = Xo [PG/ijP---~-s)No: + ok,q y.,

(1.7)

As an analogue to the osculating variety of a curve C we define the totally indicatrix variety of a vector field I/with respect to the curve C as that determined by the vector field V and its derived vector in Vm with respect to C. Further let us consider a set of (m - n) congruences of curves one curve of each of which passes through each point of In. Let 2,~ be the unit vector in the direction of a curve of the congruence which is in general not normal to V, and therefore may be specified by i • 2 a, / = t,/y,~ + ~,

(1.8)

a co~/N~/

o

where tr/i and c,,/ are parameters. It is easy to verify that [2] i j gijtr/t~/ + ~

(1.9)

2 cot /= 1

o

1.10

cos Oo,/= co,~

and

(1.11)

cos ~r/ = gijt,/P

j

where Oo,/ and ~,/are the inclinations of the vector 2, to the vectors No~ and V respectively. 2. Union curvature of a vector field. If we consider an ndicatrix in V m with the above direction ,t~/, then the condition that it is an indicatrix variety of the vector field V requires 2~/= u,/y,~pi + w,/T ~

(2.1)

where u~/ and w~/ are parameters to be determined. From (1.7), (1.8) and (2.1) it follows that i

Ct

CI

-~-

•

l

t~/y,~ + ~o cot~No~ ur/y.~p + (flo/ijpi dnY \ ~ + vkaq iY,~] _~_s )No/ and therefore we obtain

(2.2/

u,/= Pd~

1965]

UNION CURVATURE OF A VECTOR FIELD IN A SUBSPACE

(2.3)

1

dx! t)°/~JP~ ds

W ~I

Co~I

17

and dx b

(2.4)

~k,q h +

C~,l

Since (2.3) implies that

. dx j 1

D'/'sP' " - ~

W~I

DalUP

C~rr/

,dx J ds

Ca~/

therefore (2.4) may also be written as e(~kn)

h

#~! = vk"qh + ( z, ~, )X~pc2v'l/2a'/=0

(2.5)

where e(vk,), the normal curvature of the vector field V with respect to C in V, of V,, and the vector tr~ are defined by

,,k~ = ~", ( D~/"bp" --~s dxb'~2 ] and h

t h

lh

a,! = Pi(t,/P - P t,/). h The vector with components #~/will be called the union curvature vector of the vector field V with respect to the curve C relative to the congruence 2~/. Thus union curves of a vector field Vrelative to a congruence 2~/are curves along which the union curvature vector #h/is a null vector. Its magnitude which is the union curvature of the vector field V with respect to C relative to the congruence 2,/is given by =

#~/#~lh

e(~k.) h ~ e(~k,) a ~ =(vkuq h+ (~,c2,)i/2aqJ (vk.qh + ( ~ , , h ] V

*2

or

(2.6)

oeu( ) = ok; +

2~k,e(~k.)

qh ,h +

vk 2 V

If e(vk,)= 0, then (2.6) gives ~k~(~) = ~k.. Hence:

h

18

R.N. KAUL

T h e union curvature of a vector field V relative to a n y congruence )t,/ with respect to asymptotic lines of the field V is the associate curvature of the vector f i e l d V. From (2.6) we find that vk,(z) vanishes if vko and ~k, are both zero. Thus: I f a curve in V, is an indicatrix as well as asymptotic line of the vector field V then it is also a union curve of the vector field V relative to a n y congruence A,/. If the congruence be one of normals to V, in which case t~/= O, it folIows from

(2.6) that Therefore:

~k.(,') = ok,.

A union curve of a vectorfield V in V, relative to any normal congruence 2,/ is also an indicatrix of the vector field E

Making use of (1.10)and (1.11)in (2.6)we obtain for the magnitude vk,(z) of the union curvature of a vector field relative to the congruence 2,/the following relation in terms of the inclinations 0 , , / a n d ~t,/ rsin%t,/- ~cos20,,/]'/2 (2.7)

~k,(t) = vko - e(vkn) [.

~os2-~,/

•

.

Taking m = n + 1, N,~/= N ' a n d the vectors of the field Vtangent to the curve C, we find that the formula (2.7) for ~k,(~) agrees with the known formula for k, given by Springer [1]. In this case we obtain cos20\ 1/2 c~0 -) .

. /sin2a--

(2.8)

ku = k s - £,~

If the congruence be one of normals to V,, then cos0 = 1, and cos~ = 0, therefore (2.8) yields the relation k, = k s which is the known result [1] that corresponding to a normal congruence, the union curves are geodesic curves. REFERENCES 1. C.E. Springer, Union curves o f a hypersurface, Canadian of Maths., 2 (1950), 457-460. 2. R. S. Mishra, Sur certaines courbes appartenant aun sous-espace Riemannien, Bull. Sei. Maths., 19 (1952), 1-8. 3. R. N. Kaul, Union curvature of a vector field, The Tensor (New Series), 7, No. 3 (1957), 185-189.

4. C. E. Weatherburn, An introduction to Riemannian geometry and tensor calculus, Cambridge University Press, 1950. 5. T. K. Pan, Normal curvature of a vecwr field, Amer. of Maths. 74 (1952), 955-966. UNIVERSITY OF CALIFORNIA BERKELEY, CAUFORNIA

THE AFFINE IMAGE OF A CONVEX BODY OF CONSTANT BREADTH BY G . D . CHAKERIAN ABSTRACT A convex body is said to have constant diagonal if and only if the main diagonal of the circumscribed boxes has constant length. It is shown that an n-dimensionalconvex body, n ~ 3, is the affine image of a body of constant breadth if and only if it has constant diagonal. Affine images of bodies of constant breadth are also characterized by the property that the orthogonal projection on each hyl~rplane is the affine image of a body of constant breadth in that hyperplane.

A convex body in n-dimensional Euclidean space E. is a compact, convex subset with non-empty interior. A convex body has "constant breadth" if the distance between parallel supporting hyperplanes is constant. A set K' is the "affine image" of K if there exists an affine transformation f: E, -* En such that f ( K ) = K'. A rectangular parallelopiped will be referred to as a "box". A convex body has "constant diagonal" if the main diagonal of the circumscribed boxes has constant length. In this paper we are concerned with characterizing affine images of bodies of constant breadth, a problem raised by S.K. Stein. Our main theorem is, THEOREM 1. In E,, n >=3, a convex body is the affine image of a body of constant breadth if and only if K has constant diagonal. We shall also prove a related theorem, THEOREM 2. In E,, n > 3, a convex body is the affine image of a body of constant breadth if and only if its orthogonal projection on each hyperplane is the affine image of a body of constant breadth in that hyperplane. The proofs of these theorems depend on the following lemmas. LEMMA 1. In E~, n >=2, a convex body is the affine image of a body of constant breadth if and only if K + ( - K) is an ellipsoid. Proof. This follows immediately from the observation that K is a body of constant breadth if and only if K + ( - K) is spherical. Received April 2, 1965.

19

20

G.D. CHAKER1AN

[March

LEMMA 2. In E,, n > 3 a convex body is an ellipsoid if and only if its orthogonal projection on, each hyperplane is an ellipsoid in that hyperplane. Proof. If K is an ellipsoid, then all its orthogonal projections are ellipsoids. The following proof of the converse is an adaptation of the p r o o f given by Siiss [3] for the case of E3. Now suppose each orthogonal projection of K is an ellipsoid. Then each supporting hyperplane of K intersects K in just one point. Denoting the points of E, by x = (xl, ".-, x,), assume that the segment joining a = (0, ..-, 0,1) to a' = (0,...,0, - 1) is a maximum diameter of K. Then the hyperplanes x, = 1 and x, = - 1 are supporting hyperplanes of K. Let K* be the orthogonal projection of K on the hyperplane xl = 0. Then K* is an ellipsoid in xi = 0. aa' is one axis of K*, since the intersections of x, = 1 and x, = - 1 with xl = 0 are supporting (n - 2)-planes of K*, orthogonal to aa' and passing through a and a ' respectively. Let H be a supporting hyperplane of K which is orthogonal to xa = 0. Then H intersects x~ = 0 in H*, where H* is a supporting (n - 2)-plane of K*. If, in particular, H is also chosen parallel to aa', then H* is parallel to aa', and hence H* n K* lies in x, = 0. It follows that H n K lies in x, = 0. But this argument could have been applied to any supporting hyperplane parallel to aa'; hence, any supporting hyperplane parallel to aa' intersects K in a point lying in x, = 0. From this it follows that the intersection o f K with x, = 0 is identical with its orthogonal projection on x, = 0, which is an ellipsoid K**. It is easy to show that K** must be centered at the origin. Now let f : E. ~ E, be an affinity which keeps aa' fixed and maps K** onto a sphere S in x, = 0 centered at the origin. Then f ( K ) intersects x, = 0 in the sphere S. Each diameter of S is orthogonal to the supporting hyperplanes o f f ( K ) through its endpoints.Also, x, = 1 and x, = - 1 are supporting hyperplanes o f f ( K ) , orthogonal to aa' and passing through a and a' respectively. Finally, all the orthogonal projections of f ( K ) are ellipsoids (here one needs to use the fact that not only the orthogonal projections, but all projections, of K are ellipsoids). In the argument above, the only property o f aa' we actually used was that the supporting hyperplanes through a and a' were orthogonal to aa'. From this it followed that the hyperplane through the origin orthogonal to aa' intersected K in an ellipsoid. Thus if bb' is any diameter of S, the same arguments can be applied to show that the hyperplane through the origin orthogonal to bb' intersects f ( K ) in an ellipsoid; moreover, this ellipsoid has aa' as an axis. It follows that every 2-plane containing aa' intersects K in an ellipse having aa' as one axis and a diameter of S as the other. Thus f ( K ) is an ellipsoid of revolution, and K is an ellipsoid. This completes the proof. LEMMA 3. In E,, n > 3, a convex body is an ellipsoid if and only if all its circumscribed boxes have their vertices on a fixed sphere.

1965]

IMAGE OF A CONVEX BODY OF CONSTANT' BREADTH

21

Proof. The sufficiency of the condition is proved, for n = 3, in [1]. We proceed to the general case by induction. Suppose K is a convex body in E~, n > 3, all of whose circumscribed boxes have their vertices on a fixed sphere S, and assume we know the lemma for Ek, 3 < k < n. Let H be any supporting hyperplane of K, and let K* be the orthogonal projection of K on H. Then any box B* in H circumscribed about K* is a face of a box B circumscribed about K. The vertices of B lie on S; hence, the vertices of B* lie on the sphere S ~ H . Thus K* is an ellipsoid in H. It follows that the orthogonal projection of K on any hyperplane is an ellipsoid, so by Lemma 2, K is an ellipsoid. The result follows, for all n, by induction. The converse, namely that all circumscribed boxes of an ellipsoid have their vertices on a fixed sphere, is a simple matter of analytic geometry. This completes the proof. Proof of Theorem 1. 1. Let S"- 1 be the unit sphere centered at the origin in E,. A "direction" in E n is a point u ~ S n - I . N ow suppose K is the affine image of a body of constant breadth, so K + ( - K) is an ellipsoid E centered at the origin. Let p(u) be the support function of E measured from the origin, and let b(u) be the breadth function (distance between parallel supporting hyperplanes orthogonal to direction u) of K. Then p(u) = b(u) for all directions u. But if u 1,u2,'", u, are any n mutually orthogonal directions, then )ST=l[p(ui)] 2 is constant, by Lemma 3. Hence, ~ ' : l[b(ui)] 2 is constant, which is precisely the condition that K have constant diagonal. Conversely, if K has constant diagonal, then ~,'~:l[b(ui)-I 2 is constant, so ~7=l[p(ui))] 2 is constant, where p(u) is the support function of K + ( - K ) . Thus all the boxes circumscribed about K + ( - K) have their vertices on a fixed sphere. By Lemma 3, K + ( - K) is an ellipsoid, so K is the affine image of a body of constant breadth. This completes the proof of the theorem. Proof of Theorem 2. For each direction u let Eu be the hyperplane through the origin orthogonal to u. Let K~ be the orthogonal projection of K on Eu. If K is the affine image of a body of constant breath, then K + ( - K) is an ellipsoid. Hence, [K + ( - K ) ] u - - K ~ + ( - Ku) is an ellipsoid in E,, so K u is the affine image of a body of constant breadth in E~. Conversely, suppose K~ is the affine image of a body of constant breadth in E~, for each u. Then [K + ( - K)]~ = K~ + ( - K~) is an ellipsoid in E, for each u. By Lemma 2, K + ( - K) is an ellipsoid, so K is the affine image of a body of constant breadth. This completes the proof.

REMARK. An interesting characterization of affine images of curves of constant breadth in E 2 is to be desired. While Theorem 1 is true in one direction in the plane case, viz. an affine image of a curve of constant breadth has constant diagonal, the converse is false. Blaschke, in [2], gives examples of centrally symmetric

22

G.D. CHAKERIAN

convex curves with constant diagonal which are not ellipses. Such a curve could not be the affine image of a curve of constant breadth, since the only centrally symmetric curve of constant breadth is the circle. REFERENCES 1. W. Blaschke, Eine Kennzeichnende Eigenschaft des Ellipsoids und eine Funktionalgleichung aufder Kugel, Berichte der Leipziger Gesellsch. d. Wiss. math.-phys. Klasse 68 (1916), 129-136. 2. W. Blaschke, Uber eine Ellipseneigenschaft und iiber gewisse Eilinien, Archiv ftir Math. und Phys. 26 (1917), 115-118. 3. W. Siiss, Eine Elementare Kennzeichnende Eigenschaft des Ellipsoids, Math. -Phys. Semesterber. 3 (1953), 57-58.

ON

CLIQUES

IN GRAPHS

BY J.W. MOON A N D L. MOSER ABSTRACT A clique is a maximal complete subgraph of a graph. The maximum number of cliques possible in a graph with n nodes is determined. Also, bounds are obtained for the number of different sizes of cliques possible in such a graph.

§1. Introduction. A graph G consists of a finite set of nodes some pairs of which are joined by a single edge. A non-empty collection C of nodes of G forms a complete graph if each node of C is joined to every other node of C. A complete graph C is said to be maximal with respect to M if C _ M and C is not contained in any other complete graph contained in M. If the complete graph C is maximal with respect to G then C forms a clique. Some time ago Erdtis and Moser raised the following questions: What is the maximum number f(n) of cliques possible in a graph with n nodes and which graphs have this many cliques? Erd6s recently answered these questions with an inductive argument. In §§2 and 3 we determine the value off(n) and characterize the extremal graphs by a different argument. In §23 and 4 we obtain bounds for g(n), the maximum number of different sizes of cliques that can occur in a graph with n nodes. It follows from these results that g(n)~ n - [log2 n]. §2. Determining the value off(n). THEOREM 1.

I f n > 2, then f(n) =

3 n/a, 4.3tn/3j-~, [ 2.3 t"/3j,

if n = 0 (mod 3); if n = 1 (rood 3); if n = 2 (mod 3).

Proof. The theorem is easily verified if 2 < n _< 4. Let G be any connected graph with at least five nodes and which contains c(G) cliques. The set of nodes joined to any particular node x of G will be denoted by F(x). Suppose there are ~(x) complete graphs contained in F(x) that are maximal with respect to G - x, the graph obtained from G by removing x and its incident edges. Also, suppose there are [3(x) complete graphs contained in F(x) that are maximal with respect to F(x) but not with respect to G - x. From these definitions it follows that X(x), Received April 7, 1965

23

24

J.W. MOON AND L. MOSER

[March

the number of cliques of G containing x, and c(G - x), the number of cliques of G - x, are given by the following equations:

(1)

z(x) = ~(x) +/~(x);

(2)

c(O - x) = c(6) - [~(x).

Suppose nodes x and y are not joined in G. Then, if G(x; y) denotes the graph obtained by removing the edges incident with x and replacing them by edges joining x to each node of F(y), it follows that (3)

c(G(x, y)) = c(G) + Z(Y) - X(x) + a(x).

To prove this, let fl(x) = fl(x, y) + if(x, y), where fl(x, y) denotes the number of complete graphs in F(x) n F(y) that are maximal with respect to G - x - y. In transforming G into G(x; y) the contribution of these complete graphs to the total number of cliques is not affected. There is a loss, however, of the cliques counted by if(x,y). In adding the edges joining x to the nodes of F(y) it is not difficult to see that a new clique is formed for each of the complete graphs counted by ~(y) and fl'(y,x). Hence,

c(G(x; y)) = c(G) - i f ( x , y) + ~t(y) + fl'(y, x) c(O) - / r ( x , y) -/~(x, y) + X(Y), c(G) + Z(Y) - Z(x) + ~(x),

using (1) and the fact that fl(x, y ) = fl(y,x). Now let G be any graph with n nodes (n > 5) and having a maximal number of cliques. A simple argument shows that G is connected and has no node joined to every other node. If nodes x and y are not joined in G then it must be that X(x) = Z(Y), for if )~(y)> Z(x), say, the graph G(x; y) would gave more cliques than G, by (3), and this would contradict the definition of G. It also follows from (3) that g(x) = 0 for all nodes x of G. For some arbitrary node x of G let a, b,-..,f be the other nodes with which x is not joined. We may replace G tl)= G by G(2)= G(a; x) without affecting the number of cliques in the graph or the properties described in the preceding paragraph. We now replace G(2) by G (a) = G (2) (b; x) and, continuing this process, we eventually obtain a graph which has the same number of cliques as G, satisfies the properties in the preceding paragraph, and in which none of the nodes x, a,b. i.,f are joined to each other but all are joined to all the remaining nodes. We may now apply this procedure with respect to some node y in F(x). By continuing to make these transformations it is clear that we will ultimately obtain a graph G* which has as many cliques as G and which has the following simple structure: The nodes of G* may be partitioned into disjoint subsets such that two

1965]

ON CLIQUES IN GRAPHS

25

nodes are joined if and only if they do not belong to the same subset. If these subsets have Jl,J2,"',Jz nodes, where Jl +J2 + "'" +Jl = n, then it follows that (4)

c(G*) =JlJ2 ""Jr

Simple calculations show that this product assumes its maximum value when as many as possible of the subsets have three nodes and the remaining ones have two or four nodes. Since G was assumed to have a maximal number of cliques and since c(G)= c(G*), it follows that f ( n ) is given by the above expressions. §3. Characterizing the extremai graphs. Let Hn denote the graphs havingf(n) cliques described at the end of the preceding section. THEOREM 2. n>2.

I f the graph G has n nodes and f ( n ) cliques then G = Hn, if

Proof. The theorem is easily verified when 2 < n < 4. Suppose G is some graph with n nodes (n > 5) and f ( n ) cliques which is not one of the graphs H.. By the preceding argument it is possible to repeatedly modify the graph G until a graph H . is obtained, without affecting the number of cliques it contains. Let G' denote the last graph in this sequence before H,. That is, G' has f ( n ) cliques and contains two nodes x and y which are not joined to each other such that G'(x; y)

Sn. Let us suppose that n = 31, in which case H~ consists of l triplets of nodes such that two nodes are joined if and only if they do not belong to the same triplet. Since x and y are not joined it follows that they belong to the same triplet of un joined nodes in H.. Let z be the third node of this triplet. If, in G', x is joined to t, of the nodes in the i'th remaining triplet of unjoined nodes, i = 1 , 2 , . . . , l - 1 , then it is not difficult to see that (5)

Z(x) = tlt2 "'" tt- 1"

NOW X(X)= Z(Y), by the earlier argument, and it is easily seen that X(Y) _- 3l- 2. Since each t~ < 3 in (5) it must be that t~ = 3 for i = 1, 2,..., 1 - 1. That is, F(x) __ G' - x - y - z. If x is joined to z in G', then c(G') = 2.31-1. But this is less than f(n), a contradiction. Hence x is joined to every node in G' except z and y. This implies that G ' = H,, by definition. The proof of the theorem may be completed by applying a similar argument in the cases when n is congruent to 1 or 2 modulo 3. §4. A lower bound for g(n). It is not difficult to construct a graph with n nodes which contains cliques of sizes 1,2,...,[½(n + 1)]. This shows that g(n) >=[½(n + 1)] for all n. When n > 26, an improved bound is given by the following result. (In what follows all logarithms are to the base two.) THEOREM 3.

g(n) >_ n - [log n] - 2[log log n] - 4.

26

J.W. MOON AND L. MOSER

[March

Proof. We temporarily restrict our attention to the case where n > 47. To establish the lower bound for g(n) we exhibit a graph L. which has cliques of at least n - [log n] -2['log log n] - 4 different sizes. Let m be the unique integer such that n = 2" + 2m + ['log m] + (l + 3), where 0 _< 1 -< 2 " + 1 + [log(m + 1)] - [log m]. When 0 _< 1 _< 2", let L, be the graph with n nodes illustrated in Figure 1, where, for convenience, we let t = [log m] + 1

and h = 2 m-1 - 2 ' - t + 1.

(The restriction that n > 47 was made to insure that h > 0.) The symbol < k > denotes a complete graph of k nodes. We refer to the nodes in the first, second, and third columns as the A, B, and C nodes, respectively; in addition, the 2"- 1 + 1 encircled B nodes will also be referred to as D nodes. The edges of the graph L, are as follows: Each A node is joined to every other A node, and similarly for the B and C nodes. Each A node a is joined to every B node not contained in a complete graph connected with a by a dotted line in the diagram. (The D nodes are encircled to indicate that they are all joined to all the A nodes except the one indicated.) Finally, each C node c is joined to every D node not contained in a complete graph connected with c by a dotted line in the diagram.



2 and N = lP(n) with 1 < p < oo. More general, by a result of Achieser and Krein (quoted in [1, p. 82] and [2,p. 112]) it follows that (15) cannot hold for rotund spaces (if m > 2): for finite dimensional rotund spaces and m > 2 the second inequality sign of (8) is strict. 4. The absolute value of a polynomial at critical points. We conclude with a simple application of Theorem 2. THEOREM 5. L e t Pro(z) = I-L=1 " (zi = 1,..., m. T h e n p~,(zo) = 0 i m p l i e s

zi), m > 2, a n d a s s u m e

I p,.(Zo) I -_ (1 -izol)(1

(16)

that

z~ II

< 1,

÷ Izol) "-1

For m > 3 and zo ~ O, l Zo I ~ t equality holds in (16) only if m-2 z o - --

In

e ~" a n d p,,(z) = (z - e i') (z + el') " - l

Proof. By the theorem of p ' ( z o ) = 0 implies Zo • H ( z l , . . . , Let now m > 3 and 0 < only if I z i ] = 1 for all i and

Izol

Gauss and Lucas [-4, section III, problem 31] z,,). Inequality (5) of Theorem 2 (for E2) gives (16). < 1 By Theorem 2 we can have equality in (16) if m - 2 of these roots coincide, say

Z 3 =

...

~_ Z m --. __ e i~.

If dim H = 2, then it follows from the Gauss-Lucas theorem that the two critical points different from - e i~ are in the interior of the triangle H and Theorem 2 excludes equality in (16). There remains thus only the one dimensional case:

BINYAMIN S C H W A R Z

38

zl = e i`, z2 = z3 . . . . . zm = - e~'; the only critical point different f r o m - e ~" is zo = ( ( m - 2 ) / m ) e ~ and in this case equality holds in (16).

For

m = 2 equality

holds in

(16)

if lz 1 [= [z2 l = 1 (zo = (Zl + zD/2). For

m > 3 the other cases o f equality are trivial: I f p.~(0) = 0 and all I z~] = 1, then both sides o f (16) equal 1, and at multiple roots on the unit circle both sides vanish. REFERENCES 1. D.F. Cudia, Rotundity, American Mathematical Society, Symposium on Convexity, cattle, Proc. Symp. Pure Math. 7 (1963). 2. M.M. Day, Normed linear spaces (2rid printing), Springer Verlag, Berlin, (1962). 3. Obreschkoff, N., L6sung tier Aufgabe No. 10, dber, Deutsch. Math.-Verein., 33 (1925), 30. 4. G. P61ya and G. Szeg/5 Aufgaben und Lehrs~itze aus der Analysis, 1, Dover, New York (1945). 5. G. Szeg6, Aufgabe No. 10, Jber. Deutsch. Math.-Verein, 32 (1923), 16. TECHNION-ISRAELINSTITUTEOF TECHNOLOGY,HAIFA

LOCAL CONVEXITY AND STARSHAPED SETS BY

F. A. VALENTINE ABSTRACT

Previously [7] we proved among other results that a closed connected set in E,, which has a unique point of local nonconvexity is starshaped. Here we characterize a fairly large class of plane sets whose points of local nonconvexity are so arranged that starshapedness follows. This theory determines as a special case the simple closed polygonal regions which are starshaped. In order to proceed simply we utilize the following notations and definitions. NOTATIONS. The interior, closure and boundary of a set S in Euclidean n-space E n are denoted by intS, clS and bd S respectively. The closed segment joining points x ~ En, y ~ E~ is denoted by xy. Set union, intersection and difference are indicated by L), ~ and ,-, respectively. The symbol conv S denotes the convex hull of the set $. The symbol 0 denotes the empty set. DEFINITION 1. A point x ~ S is a point of local convexity of S if there exists a neighborhood N of x such that N n S is convex; otherwise x is called a point of local nonconvexity. The set of all points of local nonconvexity of S is denoted by Q. DEFINITION 2. A set S is starshaped with respect to a point p if px c S for all points x e S. DF2INITION 3. The two closed rays of a line L which have only a point x ~ L in c o m m o n are called complementary rays. I f R(x) is a ray with endpoint x, its complementary ray is denoted by R'(x). DEFINITION 4. A ray R(x) with endpoint x e bd S is an external ray of support to the int S if R(x) n i n t S = 0. DEFINITION 5. I f X is a boundary point of a set S, then K(x) is the union of all the external rays of support to int S at x. A boundary point x e S is called a onesided point of external support of int S if

K(x)

O,

and if K(x) lies in a d o s e d half-space which contains x in its boundary. The set K(x) is called an external cone of support. The union of all the complementary rays R'(x) where R ( x ) c K(x) is denoted by

r'(x). Received May 4, 1965 The preparation of this paper was supported in part by NSF Grant GP-1988. 39

40

F.A. VALENTINE

[March

The following theorem is one of Krasnoselskii type [2], and it is also related to results developed by Eugene Robkin in his thesis 15]. For another kind of theorem of Krasnoselskii type for polygonal regions see Molnar [4]. THEOREM 1. Suppose S c E 2 is a bounded plane set which is the closure of an open connected set. Then S is starshaped if and only if both of the following conditions hold. (a) Each point of local nonconvexity x e S has a nonempty cone of external support K(x) to intS at x. (b) I f xl,x2,x3 are three points of local nonconvexity of S (they need not be distinct) which are also one-sided points of external support to int S, then there exist three external rays of support R(xl),R(x2),R(x3) to int S at xl,x2, x3 respectively whose corresponding complementary rays R'(xO, R'(xz),R'(x3) are concurrent and meet in S, so that (1)

S ~ R'(Xl) N R'(x2) N R'(x3) ~ 0.

Proof. We will prove that conditions (a) and (b) imply that S is starshaped. Let Q1 denote the set of all points of local nonconvexity of S which are also onesided points of external support to int S (see Definitions 1 and 5). First, observe that if S is convex then Q1 = 0; however, in this case S is also starshaped, being convex. Hence, suppose S is not convex. Since int S ~ 0, when int S is not convex a theorem of Leja and Wilkosz [3] implies that a point q of local nonconvexity of S exists which is the midpoint of the bounding diameter of a closed semicircular region which, except for q, lies in int S. Clearly such a point q is in Qt, so that QI ~ 0. Now define C(x) as follows,

C(x) - ( c o n v K'(x)) n conv S where x E QI (see Definition 5). Since conv K'(x) is closed, and since S is compact, the set C(x) is compact. By hypothesis, every three of fewer members of the collection of compact convex sets

{C(x),xe 01} have at least one point in common. Helly's Theorem [1] then implies that (2)

M-

(']

C(x) # O.

xEQI

Let p 6 M. We will prove that S is starshaped relative to p. Firstly, if pC S, then since S is compact and connected there exists a cross-cut of the complement of S, say uv, such that (3)

u e S, v e S, S n (uv ~ u ~ v) = 0, (p, u, v) are collinear.

Secondly, if p e S and if there exists a point y ~ S such that

1965]

LOCAL CONVEXITY AND STARSHAPED SETS

(4)

41

p y q~ S

there again exists a cross-cut of the complement of S, uv, such that (3) holds. Hence, to prove p y ~ S for each y e S, we prove that (3) cannot hold. Suppose (3) does hold. Since S is closed and connected, the segment uv divides a component of the complement of S, say K, such that K .,. uv contains at least one bounded component. Let K 1 be such a bounded component of K .~ uv. This component K 1 of K ~ uv abuts uv from one side of uv. Let H be that closed half-plane whose boundary contains uv and in which K t abuts uv. Consider the set K* - cl conv(H n K1). Since K* is a compact convex set with int K* ~ 0, and since int K* c H, there must exist a point x e bd K *

and a line of support L to K* such that (5)

Lt~ K* = x, x ~ S

and such that p and uv lie in the same open half-space bounded by L. Since u e S, v e S, u # v, since S = cl int S, and since int S is open and connected, for each e > 0 there exists points u 1 ~ int S, vl~ int S such that lIul-ul]

< e, ] l v l - v i i < e

and such that a polygonal path P ( u ~ , v l ) exists in int S joining u 1 and v i. Now, clearly x is a point of local nonconvexity of S otherwise the convexity of S r~ N for some neighborhood N of x would violate condition (5). Furthermore, one can choose a bounded component Ks of K .,. uv and an e. > 0 (above) so that K ( x ) must be in the closed half space bounded by L which contains p and uv, because K ( x ) N P(Ul,Vl) = O.

However, in this case, no ray

R(x) in

conv

K(x) exists

whose complementary ray

R ' ( x ) contains p. This, however contradicts (2), since p E M. Hence, we have

arrived at a contradiction. Therefore, S is starshaped. Since the converse situation is trivial, we have proved Theorem 1. Theorem 1 has an interesting form when the boundary of S is a simple closed polygon in E2. To state it we first note the following. DrrINmON 6. A vertex x of a simple closed polygon P is called reentrant if x is a point o f local nonconvexity of the closed bounded polygonal set whose boundary is P. TrmOR~M 2. L e t S be a bounded closed set in E2 whose b o u n d a r y is a s i m p l e closed polygon. S u p p o s e that f o r each three r e e n t r a n t vertices o f bd S, x t, x 2, x a

42

F . A . VALENTINE

there exist three e x t e r n a l r a y s o f support at x l , x z , x 3 respectively to int S whose corresponding c o m p l e m e n t a r y r a y s are concurrent (see (1)) and meet in S. T h e n S is starshaped. Proof.

T h e o r e m 2 follows f r o m T h e o r e m 1 since c o n d i t i o n s (a) a n d (b) a r e

a u t o m a t i c a l l y b o t h satisfied. BIBLIOGRAPHY 1. Math. 2. 3. Math. 4. 5. 6. 7.

E. Helly, Uber Mengen konvexer KOrper mit gemeinshafilichen Punkten, Jber. Deutsch. Verein 32 (1923), 175-176. M.A. Krasnoselskii, Sur un critdre pour qu' un domaine soit dtoile, Math. Sb. (61) 19 (1946) F. Leja and W. Wilkosz, Sur une propridtd des domaines concaves, Ann. Soc. Polon. 2 (1924), 222-224. J. Molnar, Uber Sternpolygone. Publ. Math. Debrecen. 5 (1958), 241-245. E.E. Robkin. Characterizations of starshaped sets, Doct. Thesis, Univ. of Calif. (1965). F. A. Valentine. Convex Sets, McGraw-Hill Book Co. (1964). F. A. Valentine. Local convexity and Ln sets. (To appear in Proc. Amer. Math. Soc.)

UNIVERSITY OF CALIFORNIA, Los ANGELES, CALIFORNIA

VARIETE SUR

UN

DE NON-SYNTHESE

GROUPE

ABELIEN

SPECTRALE

LOCALEMENT

COMPACT

PAR

M. FIL1PP1 RESUME

On demontre quelques r6sultats concemant les ensembles de synth~se ou de r~solution spectrale dam les groupes abeliem localement compacts.

§1. Introduction. Soit F un groupe localement compact, la loi de groupe de F sera not6e additivement, et soit G son dual (groupe des caract6res g de 1"). Un caract~re de F est une fonction continue born6e sur F, et nous noterons, par abus de langage, de la m~me fa¢on un 616ment de G e t la classe dans L°°(F) de la fonction sur F qu'il d6finit. Soit V un sous-espace faiblement ferm6 invariant par translation de L~°(F), nous appellerons spectre de V l'ensemble a(V) des 616ments de G qu'il contient. Le probl~me de la synth6se spectrale est le suivant: notant par Vt la sous-vari6t6 de L°°(F), faiblement ferm6e, invariante par translation, engendr6e par les caract~res appartenants a or(V), V1 est la plus petite sous-vari6t6 de spectre a(V). Nous dirons que Vest une varidtd de synthdse spectrale si V = V~. Alors "la synth~se est possible" pour tout 616ment de V, c'est ~ dire que tout 616ment de V peut &re approch6 dans L°°(F) par des combinaisons lin6aires des caract~res appartenant a(v). Une caract6risation des vari6t6s de non-synth~s¢ spectrale semble diflicilement abordable; dans [2] P. Malliavin a montr6 l'existence des vari6t6s de non-synth~se sur un groupe ab61ien localement compact, non compact (la synth~se sur un groupe compact &ant toujours possible). On est ainsi amen5 h travailler sur une notion plus restrictive que celle de synth~se spectrale: on appeUe "varidtd de rdsolution spectrale'" une vari6t6 faiblement ferm6e V, invariante par translation, telle que toute sous-vari&6 de V ferm6e, invariante par translation, soit de synth~se spectrale.

It

Notations. Nous noterons comme d'habitude tl~ et 11"'" II1 les normes dans L~°(F) et L'(F) resp., MI(F) notera l'alg~bre des mesures born6es sur F, et nous noterons encore II "'" I11 la norme darts MS(F). Le produit de convolution dans le groupe F, sera not6 par un simple point, Received May 8, 1965.

43

44

M. FILIPP1

[March

et nous 6crirons 6galement e"a pour l'exponentielle de convolution de a dans F, e t a "2 pour le carr6 de convolution de a. Le produit de convolution dans le groupe additif des r6els sera, lui, not~ par une 6toile ( * ). Enfin, soit (I) un 61~ment de L~°(F), nous noterons ((I)) la sous-vari~t6 f e r m ~ engendr6e par (I) et ses translat6s. 1.1. Uaieit6 et r6salation speetrale. Soit V une vari6t6 ferm~e invariante par translation dans L~°(F), nous 6tudierons la synth~se spectrale dans cette varlet6 en relation avec la d6croissance gt l'infini des 616ments de V. DI~FINITION 1. Nous appellerons L~ (F) l'espace des fonctions de L°°(F) qui "tendent vers 0 ~t l'infini", c'est h dire l'espace des fonctions (I)~ L°°(F) telles que quel que soit e > 0 donn6, il existe un compact K de F, tel que, si hx note la fonction caract6ristique de K, 11(I) (1 - hr)ll ~o< ~. Nous aurons alors un th6or6me qui relie la notion de r6solution spectrale celle, mieux connue, "d'ensemble d'unicit6". Rappelons que, soit F u n groupe localement compact et G son dual, on appelle ensemble d'unicit6 un ensemble ferm6 E c G, tel que si (I) e L~(F), et que le spectre de ((I)) est dans E, alors (I) est nulle. Nous dirons qu'un ensemble ferm6 E c G, est de r6solution spectrale si il n'est le spectre que d'une vari6t6 de L°°(F), et que cette vari6t6 est de r6solution spectrale:

THI~ORf~ME1. Duns un groupe localement compact G, tout ensemble de rdsolution spectrale est un ensemble d' unicitd. Ce r6sultat a 6t6 ddmontr6 par P. Malliavin dans [3], dans le cas off le groupe G est le tore ~t une dimension C. 1.2. D6eroissanee h l'intini. Pour 6tudier la d6croissance 616ment de L°°(F), nous utiliserons les notions suivantes:

~ l'infini

d'un

D~F1NmON 2. On appelle "suite d'unit6s approch6es de/.1(1) '' une suite ~ de fonctions de LI(F), de normes 1, positives, telles que pour toute fonction sur F, a, support compact et continue, j'ra~ld~ (d~ mesure de Haar sur F) tende vers a(0) quand i tend vers l'infini (0 6tant l'616ment neutre de F). Nous savons que de telles suites existent si et seulement si F a une base d6nombrable de voisinage de l'616ment neutre. Etant donn~e une suite d'unit~s approch~es, ~ une fonction • e L®(F), et ~ tout > 0, on associe les ensembles (1.1)

A , [ ¢ , {~o,}] = {7 e r/lim sup (¢. 9,)(~) > ,}

(1.2)

B~[O, (~,}] = A,[¢, {T,}] - A,[O, {91}].

1965]

VARIETE DE NON-SYNTHESE SPECTRALE

45

Une autre mani6re canonique, mais moins fine, d'6tudier la d6croissance /~ l'infini de • est: 6tant donn6e une base d6nombrable ~ de voisinages ouverts de l'616ment neutre de V, on pose: dO(y) = lim sup [I O(x)hv(x - 7)[1~o V¢~ll

(ot~ hv est la fonction caract6ristique du voisinage ouvert V). Nous consid~rerons alors les ensembles:

(1.1')

.4,(O) = {7 E F/dO(y) > e}

(1.2')

/~,(O) = L ( O ) - L ( O ) .

Remarquons que si ia fonction • est continue ces deux notions coincident, (c'est ~ dire que A~(O, {%}) = .4~(O)), et que les 616ments de L~(F) sont les fontions • de L~(F), dont les A~(O,(%}) et les ,~'~(O) sont relativement compacts. 1.3. Enonc6 ties r6suitats. PROPRII~TI~ (P). Un ensemble B darts F poss~de la propri6t6 (P), si, quel que soit le nombre entier N, et quel que soit un nombre fini d'616ments Yi de F, on peut trouver un 61~ment y e F dont tousles multiples non nuls d'ordre inf6rieur en valeur absolue ~t N n'appartiennent pas ~ Wi(B + y~).

TrI~,ORi~ME 2. Soit F un groupe localement compact, ayant une base d(nombrable de voisinages de l'origine, soit • ~ L°°(F); supposons qu'il existe une suite d'unitds approchdes % telles que les ensembles Be(O,{9i}) possddent pour tout e > 0 la propridtd (P), alors (0) n'est pas de rdsolution spectrale. THI~ORI~ME 2'. Soit F un groupe localement compact, ayant une base ddnombrable de voisinages de l'origine, soit OEL°°(F); supposons que pour tout e > O, les ensembles B,(O) vdrifient la propridtd (P), alors (0) n'est pas de rdsolution spectrale. THI~OR[ME 3. Soit dans un groupe localement compact F, une varidtd V fermde invariante par translation de L°°(F), telle que V ~ L o ( F ) # {0} alors V n'est pas de rdsolution spectrale. Le th6or~me 1 6nonc6 plus haut n'est, bien entendu, que la forme (un peu moins pr6cise) que prend le th6or~me 3 si on le traduit, par dualit6, en termes d'ensembles dans le dual G de F (en regardant les spectres de nos vari6t6s). C'est sous ia forme du th6or~me 3 qu'il sera d6montr6. THi~OR~.ME 4. Considdrons le groupe Z"; soit O~L~°(Z ") telle que les ensembles ~ ( 0 ) soient pour tout e > 0 de "densitd in/drieure" nulle, alors (0) n'est pas de rdsolution spectrale.

46

M. FILIPPI

[March

Rappelons que si B c Z ~, on appelle "densit6 inf&ieure" de B la limite inf&ieure du rapport [B N K p l l -~ o~ est le cube des 616ments de coordonn6es, dans Z ~, toutes inf6rieures h pen valeur absolue, et }A I le nombre d'616ments de A c Z'. Enfin nous d6montrerons, comme corollaire du th6or~me 4, le THI~OR~ME 5. Dans le tore d u n e dimension C, l'ensemble de Cantor n'est pas de rdsolution spectrale. Ce r6sultat (qui a 6t6 montr6 ind6pendamment par Kahane et Katznelson dans [1]) donne alors un exemple d'ensemble d'unicit6 (l'ensemble de Cantor 6tant d'unicit6, cf. 15]) qui n'est pas de r6solution spectrale.

§2. D6monstration du th6or~me 2. La d6monstration de ce th6or~me se fera en deux temps: - - un premier lemme montrera que si il existe une fonction ¢I)E L~°(F)et une fonction a ~ U ( F ) telles que + soit int6grable (en la variable r6elle u), alors (~) n'est pas de r6solution spectrale. - - nous construirons ensuite explicitement, lorsque • v&ifie les hypotheses du th6or~me 1, une fonction a ~ L~(F), qui avec • v6rifie les hypotheses du lemme pr6c6dent.

11 (1 lul)

II

2.1. LEMME. 1. Soit F un groupe localement compact, supposons qu'il existe une mesure bornde a ~ MI(F), et une fonction dp ~ L~(F), telles que (2.1)

f

+oo

lie "s... --00

* I1 (1 + lul)du < + oo

alors il est possible de trouver une constante c r~elle, telle qu' en posant b = a + c6 (6 ~tant la mesure de Dirac ?t l'dldment neutre d e F), la fonction de L°°(F) ¢ + ~ o / "iub qs = J-~ote • O)udu vdrifie

(2.2) (2.3)

b . ~P ¢: 0 b.b.'I'

= 0

(dp) n' est pas de rdsolution spectrale.

D~mostration. Remarquons d'abord que la relation (2.1)est aussi v&ifi~ en rempla~ant a par b, car e "~'b= el'Ce "l', et el'%st un nombre de module 1. Soit F(x) une fonction de la forme (2.4)

F(x) = P(x)e -x2,

ofa P e s t un polyn6me; alors F(u) (transform6e de Fourier de F(x)) d~roit

1965]

VARIETE DE NON-SYNTHESE SPECTRALE

47

/t l'infini comme ]u [%1" n &ant le degr6 de P). Cette fonction &ant enti~re, nous pouvons d6finir la mesure born6e F(b). D'autre part, la fonction de R /t valeur darts l'espace de Banach MI(F), e "iubl~(u) est continue et de norme int6grable ([le'~Ublll <el~lllbll~), elle est done int6grable; soit I son int6grale, et g un caract6re de F, on a:

(1,g) =

(e "Ub,g)P(u)du =

e

--

- - QO

qui est 6gal d'apr~s la formule d'inversion de Fourier h 2zF((b, g)). Nous avons d0nc:

F(b)= l/2n f_~e "b P(u)du

(2.5)

u e .l~b. • est une fonction /t valeur vectorielle dans L°°(F) continue et de norme int6grable, done int6grable dans L°°(F), soit ~,+~ q / = | (kl,b .O)udu,

(2.6)

ona:

J - - - O0

F(b) " • = 1/2n

"iubl~(U)du

(d ub " ¢)t dt. co

Or la fonction de R 2 dans L°°(F), (u, t) ~ e "ttu+t)b • ¢, est continue et sa norme est major6e par le produitl[e'"bl],l]~ ''b "O lifo, done ~(u, O e(u)t est int6grable, et d'apr~s Lebesgues-Fubini on a:

F(b) " W = 1/2nL2 (e "'`u+°b • ~ ) P ( u ) t d u d t soit en posant v = u + t et en appliquant encore Fubini

F(b)" • =

2.7) f(O

~ ) ( P * f)(v)dv

= v. --X2

1) Prenons F(x) = xe , P(u) = u e-"'/4( ~ ) et son produit de convolution avec la fonetion f est une constante ~gale ~ 27r. Done

b" e "-~" • =

(e "~b .O)du, oo

soit:

b'e'-b~'~

= f ~ " ~ (e"'u"'O)e d u . _

Mais e "v'°- • &ant une fonction continue non nulle ~ l'origine ( 4 &ant suppos6e non nulle), elle ne peut pas ~tre orthogonale h toutes les fonctions e ~u~pour c r6el

48

M. F I L I P P [

[March

quelconque, d o n c il est possible de trouver c tel que b • e "-b2 • tF ne soit pas nul ce qui entraine: (2.2) b . • # 0. 2) Prenons maintenant F ( x ) = x2e - ' ' , sa transform6e de Fourier est r ( u ) = x/n-72(u2/4 - 1)e-"21+ d o n t le produit de convolution avec f est nul, d o n e b • b • e "-b2 • qJ est nul, et en convolant par e "b2, on a: (2.3)

b.b.

W = 0.

3) b • W 6tant diff6rent de z~ro, il existe une fonction ~ e L I ( F ) telle que • b • W # 0; par contre (2.3) entraine que l'id~al engendr6 par ~. b • ~ • b dans L~(F) est orthogonal b. W, done que les caraet~res de F contenus dans (W) sont o r t h o g o n a u x ~ 8 • b • e • b e t d o n c ~ e • b: ceci entraine alors que ces caract~res ne peuvent pas engendrer (qa). Enfin tF appartenant/t (q~), (~F) est inclus dans (q)) et:

(oo) n' est pas de r~solution spectrale. 2.2. LEt,rME 2. Soit

• un Jldment de L°°(F), vdrifiant les hypothdses thdordme 2, alors il existe une mesure born& a sur F, telle que:

f T,e +

++

D6monstration. N o u s allons construire cette mesure a en choisissant venablement une suite (?k) d'616ments de F, et en prenant (2.8)

du

con-

a = ~ k-2(1/2i)(3~k -- cS_ak) k

ofl si 7 e F, 6~ note la mesure de Dirac au point ?. N o u s serons amen6s, p o u r v6rifier que notre mesure a satisfait (2.1)/t utiliser des • e n t de e .t,/2) tea-e -a ) en s o m m e de majorations des coefficients du developpem mesures discr~tes de masses 1, c ' e s t / t dire les coefficients du d6veloppement en s~rie de Fourier de e +"~r, , que nous allons d o n n e r maintenant.

2.2.1. Majorations de fonetion de Bessel. Soit ~, un 616ment de F, nous noterons par r(7) l'ordre de ~,, c'est ~ dire le plus petit entier positif n tel que n • ? = 0. La fonetion e ~'jm~r'~> se d6veloppe en fonction uniquement des caraet~res de G appartenant au sous-groupe engendr~ par 7, et dont les coefficients de ce d6veloppement se calculent en se pla~ant dans le sous-groupe engendr6 par ?, art lieu de F, et son dual au lieu de G, ils ne d6pendent d o n e que de l'ordre de 7. N o u s avons les d6veloppements suivants: (2.9)

e +'°~ 3

Les d6riv6es ~ l'origine de Po(u) et P 2 ( u ) sont nulles, la d6riv6e seconde en u = 0 de Po(u) est - 1, et celle de P2(u) est 1/4; commelPo(O) I+ XI~I _ 21P~(O)I= 1, il existe un nombre ~ < 1, et un nombre ¢, 0 < ~ < ½Go tels que: (2.17)

I P , , . , ( u ) l < ~ p o u r r > 2 , mquelconque, e t ~ < l u l < 2 ¢ .

2.2.2. Estim~e de normes par r6gularisation.

Soit (~3 la suite d'unit6s approch6es de L~(F), nous devrons d6duire une majoration de normes de certaines expressions d6pendant de ~,/L partir de majorations sur les normes d'expressions analogues oh q) est remplac6 par l'une de ses r6gularis6s q~. V~ (q~- V~ sera toujours not6 dans la suite par ~3; c'est pourquoi n ous d6montrerons ici le lemme suivant: LEMME 3. Soit (~i) une suite d'unitds approchdes de U(F), soit ~ e L ~ ( F ) et supposons queen tout point ~ de F on ait lim a lors

supl.,(r~ I ___1

I1"~ II-- 1

D6monstration. Soit ~ une fonction continue d support compact K c F, cp~• converge, en norme dans U(F), vers ~: en effet, soit V un voisinage compact de l'616ment neutre 0, et soit 9,v les restrictions ~t V des fonctions 9,, II ~,- ~,. I1~ tend vers z6ro quand i tend vers l'infini, doric (9~ - 9~v) " a converge vers 0 darts LI(F), et 9~ir. • ~ converge dans LI(F ) en norme vers ~({9~v " ~} ~tant un ensemble de fonctions uniform6ment 6quicontinues ~ support dans un compact fixe, convergeant ponctuellement vers ~). La norme dans L°°(F) de ¢ est la borne sup6riere pour toutes les foctions continues ~ support compact de norme 1 dans L~(F) de [ ( ~ , , c ) l ; de plus ( ¢ , ~ ) est la limite de (¢~,~), quand i tend vers l'infini, car

VARIETE DE NON-SYNTHESE SPECTRALE

(O,,ct) = (O,- ~)(0) = (O. ~,. ~)(0)

51

(o/l ~ d6signe ~(y) = ct( - y))

et Oi " 0t converge dans L1(F) vers ~. Notons alors Sp(e) l'ensemble des y ~ F tels que quel que soit i > p; < 1+ e (e > 0 arbitrairement donnO. La r6union des Sp(e) est tout F. Soit une fonction continue h support compact de norme 1 dans L~(F), la mesure de K c3(S~,(e) tendant vers 0 quand p tend vers l'infini, il existe Po tel que l'integrale de ~ sur Spo(t) soit inf6rieure h e. Nous avons:

=(

+( ,J(Spo(~))

J Spo( O

o~ la premiere int6grale peut ~tre major6e, en module par le sup de O~ sur Spo et la deuxi~me p a r e multipli6e par t[ O,{I°o (qui est inf6rieure it O o0; doric si i>->_po, ( O i , ~ ) [ < l + e ( l + • oo). Comme ( O i , ~ ) t e n d vers (O,~), nous avons (O,~) < l , e t d o n c • ~ < 1 . 2.2.3. Construction de la suite (Yk)" Soit (t/k,~) une double suite de nombres positifs tels que: 2

(2.18)

r I (1 + qk.q) - 1 < e - ' k>q

nous savons que quel que soit r, fini ou infini, il existe (cf. (2.15)) pour tout q, une suite (dq(k)) d'entiers croissants tels que: (2.19)

~ Iral>dq(k)

[p.,,(u) I < qk,,

pour [u [ < q, et quel que soit r,

nous prendrons les entiers dq(k) croissant en fonction de q 6galement. Soit alors e(k) = It • tloo/8~. ]-Ii__ io, la norme dans U(F) de q~ restreint au compl6mentaire de W soit inf6rieure h e/114)1[o~; comme ),~ A2~ il existe un i > i o tel que 19i' 4) 1(?) > 2e, et 91 " • 6tant continue il existe une fonction continue ~t ~t support compact inclu dans W, de norme 1 dans LI(F), telle que I ~ " ~0~• (I)](7) > 2~. Alors (la restriction de ~i au compl6mentaire de W 6tant de norme inf6rieure h e/II 4)II ~o)la fonction fl, produit de convolution de et de la restriction 9~ ~ W, est continue, a son support dans V - 7 , et on a I fl" 4)[" (7) > e, ce qui entraine que II4)v IIoo > ~. Ceci &ant vrai quel que soit re voisinage V de ~, d4)(7) > e. §3. D6monstration du th6or6me 3. Soit 4) un 616ment de L~(F), pour tout 8 > 0 donn6 nous noterons A~(4)) le plus petit ferm6 tel que la norme de • restreint au compl6mentaire de A~(4)) soit inf6rieure ~ 8. Quand le groupe F a une base d6nombrable de voisinage de l'origine, les ensembles ainsi d6finis sont les .~'~(4)). Nous poserons B~(4))=A~(4))- A~(4)), et la propri6t6 de d'appartenir Lo(F) se traduit par le fait que les ensembles A~(4)) et B~(4)) sont compacts. Nous d6montrerons alors un lemme, qui, avec le th~or6me 2' d~montre le th6or6me 3 darts le cas of1 F a une base d6nombrable de voisinage de l'616ment neutre. 3.1. LEMME4. Dans un groupe localement compact non compact F, tout compact poss~de la propridtd (P). Nous allons utiliser le r6sultat classique suivant: Soit F un groupe locaIement non compact ab61ien, et soit U un voisinage compact sym6trique de l'unit6. Soit F' le sous-groupe engendr6 par U, alors F' contient un sous-groupe discret D engendr6 par un nombre fini d '616ments tel que F'/D soit compact. Nous allons traiter les deux cas suivants: a) F contient un 616ment ~'o d'ordre infini qui engendre un sous-groupe discret. b) Si on n'est pas darts les conditions du a), D est un groupe compact (engendr~ par un nombre fini d'616ments d'ordre fini), donc aussi F'. F' 6tant engendr6 par u n

1965]

VARIETE DE NON-SYNTHESE SPECTRALE

57

voisinage de l'origine, il est ouvert donc F/F' est un groupe discret, soit H, dont tousles 616ments sont d'ordre fini. La r6union d'un nombre fini de compacts 6tant compact, nous allons d6montrer que pour un compact K et un nombre entier positif arbitraire N i l existe un 616ment ? de F tel que tous ses multiples non nuls d'ordre n avec [n I < N, soient hors de K. a) Soit (Yo) le groupe ~o " Z engendr6 par ~o; ou bien K ne rencontre pas (Yo) et le probl~me est r6solu en prenant y = Yo, ou bien K rencontre (Yo) et K n (~'o) est compact done fini; soit alors A l e plus grand des entiers tels que A. ~o E K n (~'o), il suffit de prendre pour y l'616ment (IAI + 1) b) l'image K ' de K par la projection F ~ F / F ' = H est compacte done finie, si t E F / F ' est tel que tous ces multiples non nuls et d'ordre ] n [ < N ne sont pas dans K', un repr6sentant y dans F de la classe t r6pond ~t la question. I1 suffit de traiter le probl6me dans le cas d ' u n groupe F discret dont tous les 616ments sont d'ordre fini. Soit K une partie finie d'un groupe discret infini; soit N u n entier positif donn6, supposons que pour tout 616ment ~ F, 3 n < N tel que ny :P 0 et n~ ~ K; en d'autres termes, soit K . l'ensemble des y ~ F tels que n y # O et n T ~ K : nous supposons que F = Un__