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E A* such that x = vw, and it is said to be a, proper left factor if v ¥= x. The relation "t> is a left factor
Words 1.2
of x" is again an order on A*; it will be denoted by v^x. This order has the fundamental property that if
then v and v' are comparable: v < t/ or t / < u. More precisely, if i?vv = v'w',
either there exists s E ^4* such that v = t/s (and then sw = w') or there exists tEA* such that v'=vt (and then w = /w'). This will be referred to as the property of equidivisibility of the free monoid. The definition of a right factor is symmetrical to that of a left factor. The reversal of a word w = ala2- - - an, a^E A, is the word w=
an--a2al.
Hence i; is a left factor of x iff v is a right factor of x. We shall also use the notation w instead of w; we may then write or all w, v E A + ,
A word w is palindrome if w = H>. A word t> E ^4* is said to be a subword of a word x E A* if
and there existy 0 , yl9...9ynEA* x=
such that yoalyla2'-anyn.
Therefore v is a subword of x if it is a sub-sequence of ;c.
1.2.
Submonoids and Morphisms
A submonoid of a monoid M is a subset N of M containing the neutral element of M and closed under the operation of M\ NN C N. Given a suf set X of the free monoid ^4*, we denote by X* the submonoid of A* generated by X. Conversely, given a submonoid P of A*, there exists a unique set X that generates P and is minimal for set-inclusion. In fact, X is the set
X=(P-\)-(P-\f
1.2 Submonoids and Morphisms
5
of the nonempty words of P that cannot be written as the product of two nonempty words of P. It is a straightforward verification that X generates P and that it is contained in any set Y C A* generating P. The set X will be referred to as the minimal generating set of P. A monoid M is said to be free if there exist an alphabet B and an isomorphism of the free monoid B* onto M. For instance, for any word wE A+ the submonoid generated by w, written w* instead of {w}*, is free. It is very important to observe that not all the submonoids of a free monoid are themselves free (see Example 1.2.2). PROPOSITION 1.2.1. Let P be a submonoid of A* and X be its minimal generating set. Then P is free iff any equality
implies n — m andxt — yi9
Ki}*, a submonoid with two generators. The properties of conjugacy in A* proved thus far can be viewed as particular cases of the properties of conjugacy in the free group on A (see Problem 1.3.1). If Card(^4) = k is finite, let us denote by \pk(n) the number of classes of conjugates of primitive words of length n on the alphabet A. If w is a word of length n and if w — zq with z primitive and n — qd9 then the number of conjugates of w is exactly d. Hence
kn=2d+k(d)9
(1.3.6)
d\n
the sum running over the divisors of n. By Mobius inversion formula (see Problem 1.3.2) this is equivalent to:
**(») = ;£ 2 M(«0*"/rf
(1-3.7)
d\n
where /A is the Mobius function defined on N - 0 as follows:
/»(»)=(-1)' if n is the product of / distinct primes and fi(") = 0
if n is divisible by a square. Proposition 1.3.1 admits the following refinement (Fine and Wilf 1965):
10
Words 1.3
PROPOSITION 1.3.5. Let x, yGA*,n\x\,m= \y\,d = gcd(n,m). If two powers xp andyq of x andy have a common left factor of length at least equal to n + m — d, then x and y are powers of the same word.
Proof. Let u be the common left factor of length n + m — d oi xp, yq. We first suppose that d — 1 and show that x and y are powers of a single letter. We may assume that n^m — 1. It will be enough to show that the first m — 1 letters of u are equal. Denote by u{i) the /th letter of u. By hypothesis, we have K(I) = K(I + / I ) ,
K/<w-l,
(1.3.8)
u(j) = u(j + ™)>
Kj2
'
The sequence of the lengths \n = \fn\ is the Fibonacci sequence. Two consecutive elements Xn and X w+1 for n>3 are relatively prime. Let gn be the left factor of/, of length Xn -2 for n > 3. Then Sn+\ ~ Jn-\Sn-2
for n > 5, as it may be verified by induction. We then have simultaneously
Therefore, for each n>5, f2n and f3n-x have a common left factor of length \n + Xw_! —2. This shows that the bound given by Proposition 1.3.5
1.4 Formal Series
is optimal. For instance,
u — abaababaaba
1.4.
Formal Series
Enumeration problems on words often lead to considering mappings of the free monoid into a ring. Such mappings may be viewed (and usefully handled) as finite or infinite linear combinations of words (see for instance Problem 1.4.2). This is the motivation for introducing the concept of a formal series. Let A' be a ring with unit; in the sequel K will be generally be the ring Z of all integers. A formal series (or series) with coefficients in K and variables in A is just a mapping of the free monoid A* into K. The set of these series is denoted by K((A)). For a series o€zK((A)) and a word wGA*, the value of a on w is denoted by (a, w) and called the coefficient of w in a; it is an element of K. For a set X C A*, we denote by X the characteristic series of X, defined by
The operations of sum and product of two series defined by: (a + T, H>> = (a, w) + (T, W) , (OT,W) =
2
O9TEK((A))
are
>,
w = uv
for any wG ^4*. These operations turn the set K((A)) into a ring. This ring has a unit that is the series 1, where 1 is the empty word. A formal series oGK((A)) such that all but a finite number of its coefficients are zero is called a,polynomial. The set K(A) of these polynomials is a subring of the ring K((A)). It is called the free (associative) ^-algebra over A (see Problem 1.4.1). For each <JG K((A)) and TG K(A), we define
This is a bilinear map of K((A))X K(A) in K. The sum may be extended to an infinite number of elements with the following restriction: A family (o^^j of series is said to be locally finite if for each wE A*, all but finitely many of the coefficients (a,., w) are zero.
12
Words 1.4
If (a / ) / G / is a locally finite family of series, the sum
is well defined since for each n>E A*, the coefficient (a, w) is the sum of a finite set of nonzero coefficients (oi9w). In particular, the family (w)^^* is locally finite, and this allows to write for any oEK((A))
or, by identifying w with w,
a— 2 (a,w)w. This is the usual notation for formal series in one variable: o=
2 ona"
with on-(o,an). Let a be a series such that (a, 1) = 0; the family ( a ' ) / > 0 is then locally finite since (o\w) = 0 for z ^ | w | + l . This allows us to define the new series
which is called the star of a. It is easy to verify the following: PROPOSITION 1.4.1. Let oG K((A)) is the unique series such that:
be such that ( r , 1) = 0. The series a*
a * ( l - a ) = ( l - a ) a * = l. Following is a list of statements relating the operations in K((A)) with the operations on the subsets of A* when K is assumed to be of characteristic zero. PROPOSITION
1.4.2. For two subsets X, Y of A*, one has
(i) letZ=XUY. Then Z = X+Y if (ii) let Z=XY. Then Z = XY iff xy = x'y'^>x = x\ y = y\ for JC, x'E y (iii) let XCA + ,andP = X*. Then P = X* iff X is a code. The proof is left to the reader as an exercise.
Problems
13
Notes The terminology used for words presents some variations in the literature. Some authors call subword what is called here factor the term subword is reserved for another use (see Chapter 6). Some call prefix or initial segment what we call a left factor. Also, the empty word is often denoted by e instead of 1. General references concerning free submonoids are Eilenberg 1974 and Lallement 1977; Proposition 1.2.3 was known to Schiitzenberger (1956) and to Cohn (1962). The defect theorem (Theorem 1.2.5) is virtually folklore; it has been proved under various forms by several authors (see Lentin 1972; Makanin 1976; Ehrenfeucht and Rozenberg 1978). The proof given here is from Berstel et al. 1979, where some generalizations are discussed. The results of Section 1.3 are also mainly common knowledge. For further references see Chapters 8 and 9. The standard reference for Section 1.4 is Eilenberg 1974.
Problems Section 1.1 1.1.1. (Levi's lemma). A monoid M is free iff there exists a morphism X of M into the monoid N of additive integers such that X~ \0) = 1M and if for any JC, y, z, /E M xy — zt
implies the existence of a uE M such that either x = zw, uy -1 or xu — z, y — ut. 1.1.2. Let A be an alphabet and A = {a\a_E A] be a copy of A. Consider in the free monoid over the set A U A the congruence generated by the relations ad — da = 1, aEA. a. Show that each word has a unique representative of minimal length, called a reduced word. _ b. Show that the quotient of (A U A)* by this congruence is a group F\ the inverse of the reduced word w is denoted by vv. c. Show that for any mapping a of A into a group G, there exists a unique morphism
sending a G ^ o n z.) d. Apply the foregoing method to show that for W— {aba} one has
(See Schutzenberger 1964; for a general reference concerning linear equations in the ring Z((A))9 see Eilenberg 1974.)
CHAPTER 2
Square-Free Words and Idempotent Semigroups
2.0.
Introduction
The investigation of words includes a series of combinatorial studies with rather surprising conclusions that can be summarized roughly by the following statement: Each sufficiently long word over a finite alphabet behaves locally in a regular fashion. That is to say, an arbitrary word, subject only to the constraint that it be sufficiently long, possesses some regularity. This claim becomes meaningful only if one specifies the kind of regularities that are intended, of course. The discovery and the analysis of these unavoidable regularities constitute a major topic in the combinatorics of words. A typical example is furnished by van der Waerden's theorem. It should not be concluded that any sufficiently long word is globally regular. On the contrary, the existence of unavoidable regularities leads to the dual question of avoidable regularities: properties not automatically shared by all sufficiently long words. For such a property there exist infinitely many words (finiteness of the alphabet is supposed) that do not satisfy it. The present chapter is devoted mainly to the study of one such property. A square is a word of the form ww, with u a nonempty word. A word contains a square if one of its factors is a square; otherwise, the word is called square-free. For instance, abcacbacbc contains the square acbacb, and abcacbabcb is square-free. The answer to the question of whether every sufficiently long word contains a square is no, provided the alphabet has at least three letters. As will be shown, the existence of infinitely many square-free words is equivalent to the existence of a square-free word that is infinite (on the right). The formalism of infinite words has the advantage of allowing concise descriptions. Furthermore, infinite iteration of a morphism is a natural and simple way to construct infinite words, and this method applies especially to the construction of infinite square-free words. We start with the investigation of a famous infinite word, called after its discoverers the word of Thue-Morse. This word contains squares, but it is 18
2.1 Preliminaries
19
cube-free and even has stronger properties. Then we turn to the study of infinite square-free words. A simple coding of the Thue-Morse infinite word gives an example of an infinite square-free word. We then establish a general result of Thue that gives other infinite square-free words. A more algebraic framework can be used for the theory of square-free words. Consider the monoid M = A*U0 obtained by adjoining a zero to the free monoid A*. Next consider the congruence over M generated by the relations
uu^O
(uGA+).
The fact that there exist infinitely many square-free words can be rephrased: The quotient monoid M / « is infinite, provided A has at least three letters. A natural analogue is to consider the free idempotent monoid, that is, the quotient of A* by the congruence generated by
uu~u
(uEA+).
We will show, in contrast to the previous result, that for each finite alphabet A, the quotient monoid v4*/~ is finite. Many results, extensions, and generalizations concerning the problems just sketched are not included in the text. They are stated as exercises or briefly mentioned in the Notes, which also contain some bibliographic remarks.
2.1.
Preliminaries
Before defining infinite words, let us fix some notations concerning distinct occurrences of a word as a factor in a given word. Let A be an alphabet, wGA + . Let u be a nonempty word having two distinct occurrences as a factor in w. Then there are words x, y, x', y'E. A* such that
w = xuy — x'uy',
x^x'.
These two occurrences of u either overlap or are consecutive or are disjoint. More precisely, we may suppose | J C | < | ; C ' | . Then three possibilities arise (see Figure 2.1). (i) |x'| > \xu\. In this case, x' — xuz for some zEA + , and w = xuzuy'. The occurrences of u are disjoint. (ii) |JC'| = |JCK|. This implies that x' — xu, and consequently w = xuuy' contains a square. The occurrences of u are adjacent. (iii) | x ' | < | x w | . The two occurrences of u are said to overlap. The following lemma gives a more precise description of this case.
20
Square-Free Words and Idempotent Semigroups 2.0
X
w
^>
-\
y
X'
u
y
(i)
u w
u (ii)
u ^\
w
u (in) Figure 2.1. Two occurrences of u in w: (i) disjoint occurrences, (ii) adjacent occurrences, (iii) overlapping occurrences.
LEMMA 2.1.1. Let w be a word; then w contains two overlapping occurrences of a word M ^ I iffw contains a factor of the form avava, with a a letter and v a word.
Proof Assume first w = xuy = x'uy\ where the occurrences of u overlap. Then |JC| < |JC'| < \xu\ < \x'u\. Consequently x' = xs,
xu — x'z,
x'u — xut
for some nonempty words s, z, t, whence u = sz = zt.
(2.1.1)
Let a be the first letter of s, and therefore also of z by Eq. (2.1.1). Set s — av, z = az'. Then by (2.1.1) u — avaz' and w — xsuy' — xavavaz'y'. Conversely, if avava is a factor of w, then u — ava clearly has two overlapping occurrences in w. • A word of the form avava, with a a letter, is said to overlap. Thus, according to the lemma, a word has two overlapping occurrences of a word iff it contains an overlapping factor. We now turn to the definition of infinite words. Let A be an alphabet. An infinite word on A is a function
a: N-+A.
2.1 Preliminaries
21
We use the following notation
and also *
=
Proof. Note that by (2.1.4) we have
and therefore p(tn) = t2nt2n+\ for n > 0. By the definition of JU, this implies (2.2.2) Formula (2.2.1) holds for n = 0. Thus let n > 0. If n = 1m, then tn = rm by (2.2.2), and d2(n) = d2(m). Thus (2.2.1) holds in this case. If n = 2m + l9 then tn-tm and rf2(AI) = 1 + ^ 2 ( w ) m o d 2 - Therefore (2.2.1) holds in this case too. •
2.2 The Infinite Words of Thue-Morse
25
The inspection of t shows that t is not square-free. However, we will prove the following: THEOREM
2.2.3. The infinite word t has no overlapping factor.
COROLLARY
2.2.4. The infinite word t is cube-free.
The proof of the theorem uses two lemmas. LEMMA
2.2.5. Let X= {ab9 ba}; ifxEX*,
then axa £ X* and bxb £ X*.
Proof By induction on |JC|. If | * | = 0 , then indeed aa,bb&X*. Let x E X*, x T^ 1 and suppose u — axa E X* (the case bxb E X* is similar). Then u — xxx2- - - xr, with * ! , . . . , x r E X\ consequently xx — ab and xr — ba. Thus u — abyba with y — x2 • • • xr_ j E X*. But now by induction x = byb is not in X*, contrary to the assumption. • 2.2.6. Let wE A +. If w has no overlapping factor, then fi(w) has no overlapping factor. LEMMA
Proof Assume that /i(w>) has an overlapping factor for some wE A*. We show that w also has an overlapping factor. By asumption, there are x, v, yE A*, cE A with /x(w) = xcvcvcy Note that |ct>ct>c| is odd, but [i(w)E X* with X— {ab, ba}: therefore |/x(w)| is even and \xy\ is odd. Thus • Either: | JC| is even, and x, cvcv, cyE X*, • Or: |JC| is odd, and JCC, vcvc, yE X*. This implies that | v \ is odd, since otherwise we get from cvcv E X* (resp. vcvcE X*) that both t>, cvc are in X*9 which contradicts Lemma 2.2.5. In the case | x \ is even, it follows that cv is in X* and w = rsst with jx(r) = x, n(s) = cv, n(t) — cy. But then s and / start with the same letter c and ssc is an overlapping factor in w. In the case |JC| is odd, similarly vcE X*, and w = rsst with /x(r) = xc, ix(s) = vc,ix(t) = y. Here r and s end with c and ess is an overlapping factor in w.
Proof of Theorem 2.2.3. Assume that t has an occurrence of an overlapping factor. Then it occurs in a left factor fik(a) for some k>0. On the other hand, since a has no overlapping factor, by iterated application of Lemma 2.2.6 no iin{a)(n ** 0) has an overlapping factor. Contradiction. •
26
Square-Free Words and Idempotent Semigroups 2.3
2.3.
Infinite Square-Free Words
The infinite word of Thue-Morse has square factors. In fact, the only square-free words over two letters a and b are a, b, ab, ba, aba, bab.
On the contrary, there exist infinite square-free words over three letters. This will now be demonstrated. As before let A — {a, b], and let B — {a, b, c}. Define a morphism 8:B*->A* by setting d(c) = a,
8(b) = ab,
S(a) = abb
For any infinite word b on B, =
8(bo)S(bt)---S(bn)--
is a well-defined infinite word on A starting with the letter a. Conversely, consider an infinite word a on A without overlapping factors and starting with a. Then a can be factored as
with each ynE{a, ab, abb) = S(B). Indeed, each a in a is followed by at most two b since bbb is overlapping, and then followed by a new a. Moreover, the factorization (2.3.1) is unique. Thus there exists a unique infinite word b on B such that 8(b) = a. THEOREM 2.3.1. Let a be an infinite word on A starting with a, and without overlapping factor, and let b be the infinite word over B such that 8(b) = a; then b is square-free.
Proof Assume the contrary. Then b contains a square, say uu. Let d be the letter following uu in one of its occurrences in b. Then 8(uud) is a factor of a. Since 8(u) = av for some vEA* and #(*/) starts with a, a contains the factor avava. Contradiction. • By applying the theorem to the Thue-Morse word t, we obtain an infinite square-free word m over the three letter alphabet B such that 8(m) = t. This infinite word is m = abcacbabcbacabcacbacabcbabcacbabcbacabcbabc- • •
2.3 Infinite Square-Free Words
27
Note that the converse of Theorem 2.3.1 is false: There are square-free infinite words b over B such that 8(b) has overlapping factors (see Problem 2.3.7). There are several alternative ways to obtain the word m. We quote just one. PROPOSITION 2.3.2. Define a morphism B* (with B = {a, b, c}) by t>y are not square-free. According to (i), ax — a — an, whence xu — uv — vy.
The first equation shows that | x \ — \ v \. In view of xu = vy, it follows that x = v. Consequently vu — uv. By a result of Chapter 1, a(a) — uv is not a primitive word and thus is not square-free. This contradicts condition (i) and proves the claim. Now we prove the theorem. Assume the conclusion is false. Then there is a shortest square-free word wE A+ such that a(w) contains a square, say a(w) = yuuz
withw^l.
Set w = axa2• • • an9 vtf = a{at) (at E A). By condition (i), one has n>4. Next y is a proper left factor of vx and z is a proper right factor of vn since w was chosen shortest. Also yu is not a left factor of vx since otherwise t>2£>3 is a
2.3 Infinite Square-Free Words
29
factor of w, hence of vx, violating condition (ii). For the same reason, uz is and a not a right factor of vn. Thus there is an index j (\<j, = y', t>• = s, whence by Eqs. (2.3.2) u = © , t y • • VJ_XVJ
= »,+,••• ©„_,*'.
a(w)
(i)
y
v,
V
Zr
00 Figure 2.2. Occurrence of uu in «(w): (i) localization of ww, (ii) double factorization of w.
30
Square-Free Words and Idempotent Semigroups 2.3
Since a(A) is a prefix code, this implies vx = t>y+,,.. .,t>y_, = vn_x, Vj = z', and since Vj — z'*^vn, we have vn = t>y. Thus w = (flj • • • fly)2 is a square. Next consider the case j>f ¥= 1. Multiplying (2.3.2) by s and z, and by y and /, gives swz = (sy')v2- • • V ^ J Z ) = ^ ^ + 1 • • • «„_!«„ yut = vxv2-
- v^^^
(yt)vJ+x-
- vn^x(z't)
(2.3.3) (2.3.4)
Consider Eq. (2.3.3) first. Then the claim can be applied to each of the v2,...9Vj_x. Consequently a2- • • a^_x is a factor of ajaJ+l' • • an_xan. Since 5 7^1,^2***^-1 i s neither a left nor a right factor of aj9...9an; thus a 2 • • •fly-1is a factor of aJ+ x • • • #„_ } and /?a2- • • aj_xq = aj+x- • aw_!
(2.3.5)
for s o m e p , q £ A*. Now consider Eq. (2.3.4). As before, aJ+x- • • an_x is a factor of #! • • •fly.,and since neither j r nor z' is the empty word,fly_x • • • «„_ j is a factor of fl2* • • fly-i- Thus paj+X' - • an_xq = a2- • • a}_x
(2.3.6)
for some^,^G^l*. By (2.3.5) and (2.3.6), ppa2- - • aj_xqq = paj+x-
• flw_i^ = fl2' * *
a
j-\
showing that p — p — q — q — \. Thus setting X =
fl2-"fly_1=fly+1-
''an_X
we have w = axxajxan
(2.3.7)
whence by (2.3.3) and (2.3.4) st = Vj = sy\
z'z — x)n — sz,
yyf—^\ — yt
Thus the word
is not square-free. By condition (i), axajan is not square-free. Therefore ax — cij or aj — an. In view of (2.3.7), w contains a square. This yields the contradiction. •
2.4 Idempotent Semigroups
31
Example. A tedious but finite computation shows that the morphism a: A* -» A* with A = (a, b, c) defined by a{ a) — abcab,
a( b) = acabcb,
a( c) = acbcacb
fulfills the two conditions of Theorem 2.3.3 and therefore is a square-free morphism.
2.4.
Idempotent Semigroups
Let A be an alphabet having at least three letters. Then there are infinitely many square-free words in A*. As already mentioned in the introduction, this fact can be rephrased as follows. Let A* U 0 be the monoid obtained by adjoining a zero to A, and consider the congruence « generated by
Each square-free word constitutes an equivalence class modulo this congruence. Consequently the quotient monoid A* U 0 / « is infinite. There is another situation where square-free words can be used. Let m, n > 2 be fixed integers and consider the congruence = over A* generated by um = u\
u<EA*.
(2.4.1)
Once more, each square-free word defines an equivalence class, and thus the monoid A*/= is infinite. In fact, this result also holds for a two-letter alphabet (Brzozowski, Culik II, and Gabrielian 1971). These considerations can be placed in the framework of the classical Burnside problem (originally, the Burnside problem was formulated for groups only, but it is easy to state for semigroups also): Is every finitely generated torsion semigroup finite'} (A torsion semigroup is a semigroup such that each element generates a finite subsemigroup.) We have just seen that the answer is negative in general, and this is due to the existence of infinitely many square-free words. For groups, the answer also is negative (see Chapter 8 in Herstein 1968). Moreover, the groups of exponent n— that is, groups where each element has exponent n—are in general infinite (see Adjan 1979). The proof uses the fact that there are infinitely many squarefree words. For another result on the Burnside problem, see Chapter 7, Section 7.3. In one special case, surprisingly, the answer is positive. Let A be an arbitrary finite alphabet, and consider the congruence ~ generated by the
32
Square-Free Words and Idempotent Semigroups 2.4
relations ww~w,
WELA*.
(2.4.2)
The quotient monoid M=A*/~ is called the free idempotent monoid on A; indeed, any element in M is idempotent (mm = m), and any finitely generated idempotent monoid is easily seen to be a quotient of a free idempotent monoid. THEOREM 2.4.1 (Green-Rees). The free idempotent monoid on A is finite and has exactly
2 (I) n (*-«- + l)2'
(2.4.3)
elements, where n — Card(A). The numbers (2.4.3) are growing very rapidly. For n — 0,1,2,3,4, they are 1,2,7,160,332381. Before starting the proof, it will be interesting to note the difference between the relations (2.4.1) and (2.4.2). For the congruence defined by (2.4.1), two distinct words can be congruent only if both contain at least one p\h power, for /? = min(m, n). On the contrary, two distinct square-free words may be congruent for ~ . Indeed, the defining relations allow introduction of squares and then dropping of other ones. We give now a nontrivial illustration of this situation by verifying that x~y with x = bacbcabc and y — bacabc. Both x and y are square-free words, and they are also equivalent. Indeed, note first that with u = abcaca, we have (boldfaced factors are those to be reduced) uy = afrcacabacabc ~ abcac&bc ~ abcabc ~ abc whence x — (bacbc)abc~ bacbcuy — vy for t) = bacbcu. Next, for r = bcabacbcacbcbac, we have xr = bacln&bcbcabacbcacbcbac ~ bacbcabacbcaek /?at ~ b&cbcsLcbcbac ~ bacbcbac ~ bacbac ~ bac whence y = bacabc ~ xrabc ~ xs with s — rabc. Finally, x~vy~ which proves the claim.
vyy ~xy~
xxs ~ xs ~ y,
2.4 Idempotent Semigroups
33
Proof of Theorem 2.4.1. Recall from Chapter 1 that for wG^*,
It is clear that x ~ y implies alph(jc) = alph(j>). First we prove the following claim: Claim (i). / / alph(j>) C alph(x), there exists u such that x ~ xyu. This is indeed clear if y — 1. Assume |>>| ^ 1, and let y — y'a with aE A. By induction, there is a word u' such that x~xy'uf. Furthermore, aE alph(jc), whence x = zaz''. Thus for u — z'y'u' xyu — zaz'y'az'y V ~ zaz'y'u' — xy'u' ~x. This proves Claim (i). For x G i + , let x' be the shortest left factor of x such that alph(V) = alph(x). Setting xf — pa for some/?E A*, aEA, we have alph(/?) = alph(jc) ~{a}. Symmetrically, the shortest right factor x" of x with alph(x") = alph(x) has the form x" — bq for some bEA,qE A* and alph(#) = alph(x) — {b}. Thus to JC there is associated a quadruple (/?, a, b, q). We write this fact x — {p,a,b, q\ and prove: Claim (ii). If x — (p,ay b, q), then x ~ pabq. Indeed let, x — pay — zbq. Since alph(j) Calph(jc) = alph(/?a), there is by (i) a word u such that pa ~ payu = xu. Since alph(/?«)Calph(^), the dual of (i) shows that there is a word v with bq ~ vpabq = vx, where x = j?a^. This implies that Jc = pabq ~ xubq = JCW for w = wfe^r and x = zbq ~ ZUJC = tx
for / = zv. Whence JC ~ tx ~ /JCJC ^ xJc ~ xxw ~ I H " - J C .
This proves (ii). In view of Claim (ii), we can show that M is finite as follows. Assume that the finiteness holds for alphabets that have fewer elements than A. If x — (/?, a, b, q\ then Card(alph(/?)) < Card(^) and Card(alph(#)) < Card(v4), thus there are only finitely many ps and qs modulo ~ . Since there are only finitely many letters, M itself is finite. In order to compute the number of elements in M, we prove the following equivalence.
34
Square-Free Words and Idempotent Semigroups 2.4
Claim (iii). Let x = (p,a,b,q) p~ p\ a — a\ b — b\ q ~ q'.
and x'± (a\ a\ b\ q')\ then x~x' iff
Suppose first that p~ p\ a = a\ b — b\ q ~ qf. Then pabq ~ p'a'b'q' and x ~ xf by (ii). Suppose now x ~ x'. One can assume that x — a/?y, x' — a(52y for some \Vords a, /?, yE yl*. We distinguish two cases. Setting x = /?ay, we have
Case 1. \a/3\>\p\.
a^ — pat,
z — ty
for some fin A + . Then x' = patfiy and alph(/>) = a l p h ( x ) - {«} = alph(x r ) — {a}. Thus by definition// = p and a' = a. Case 2. \afi\ 3) such that ej^x) > k — 2 for all x occurring twice in a. F. Dejean (1972) improves this inequality. She constructs an infinite word a over three letters such that
for all factors x occurring twice in JC. She also shows that this lower bound is optimal. Pansiot, in a forthcoming paper, handles the case of four letters. For more than four letters, the sharp value of the lower bound remains unknown. Square-free morphisms and more generally &th-power-free morphisms are investigated in Bean, Ehrenfeucht and McNulty 1979. Characterizations of square-free morphisms are given in Berstel 1979 and Crochemore 1982. Bean et al. introduce the very interesting concept of so-called avoidable patterns, which are described as follows: Let E and A be two alphabets. For easier understanding, E will be called the pattern alphabet, a word in E+ is & pattern. Let w — exe2- • • en {et G E) be a pattern. A word u in A+ is a substitution instance of w iff there is a nonerasing morphism X: E*^>A* such that u = \(w). Equivalently, u — xxx2'"Xn with xu...9xn€zA+ and with xx = Xj whenever e^ — ej. Setting
36
Square-Free Words and Idempotent Semigroups 2.4
for example E = {e}9 A = {a, b, c}, the word u = abcabc is a substitution instance of ee. A word u in A+ avoids the pattern w in E+ iff no factor of u is a substitution instance of w. Thus for example uEA+ avoids the pattern ee iff u is square-free, and u avoids ee'ee'e iff u has no overlapping factor. Given a pattern w in E+, w is called avoidable on A if there exist infinitely many words uinA + that avoid w. The existence of infinite square-free words, and infinite words without overlapping factor can be rephrased as follows: The word ee is avoidable on a three-letter alphabet, the word ee'ee'e is avoidable on a two-letter alphabet. This formulation, of course, raises the question of the structure of avoidable patterns. Among the results of the paper of Bean et al., we report the following: Let n = Card E\ then there is a finite alphabet A such that every pattern w with |w| > 2n is avoidable on A. Another interesting extension of square-freeness is abelian square-freeness, also called strong nonrepetitivity. An abelian square is a word MM', such that M' is a rearrangement of M, that is |u\ a — \u'\a for each letter a. A word is strongly nonrepetitive if it contains no factor that is an abelian square. Calculation shows that over three letters, every word of length ^ 8 has an abelian square. On the other hand, Pleasants (1970) has shown that there is an infinite strongly nonrepetitive word over five letters. This improves considerable the previously known bound of twenty five letters given by Evdokomov in 1968. The case of four letters is still open. For related results, see Justin 1972, T. C. Brown 1971, and Dekking 1979. Concerning idempotent semigroups, Theorem 2.4.1 is a special case of a more general result also due to Green and Rees (1952). Let r ^ l be an integer. Then the two following conditions are equivalent: (i) Any finitely generated group G such that xr — 1 for all x in G is finite, (ii) Any finitely generated monoid M such that xr+ ] = x for all x in M is finite. The case considered in Theorem 2.4.1 is r = 1, and in this case the group G is trivially finite. For a proof of the theorem, see Green and Rees 1952 or Lallement 1979. Note that there are integers r such that condition (i), and consequently (ii), does not hold; r = 665 is such an integer (see Adian 1979). Moreover, Theorem 2.4.1 was generalized by Simon (1980) who proved the result that for a finitely generated semigroup S the following three conditions are equivalent: (i) S is finite. (ii) S has only finitely many nonidempotent elements, (iii) There exists an integer m such that for each sequence (sl9...9sm)inS there exist i, j (i < j) such that st- • • Sj is idempotent.
Problems
37
Problems Section 2.1 2.1.1. Let P be a property of words of A* such that I={w\P(w)} is a two-sided ideal. Each infinite word on A has a factor in / iff A* — I is finite {Hint: Apply Lemma 2.1.2 to (not P).) (See Justin 1972.) Section 2.2 2.2.1. Assume that www is a left factor of f. Then for some n > 0, | w| = 2" or | w| = 3.2" and | w| is a multiple of 2". If
> be the formal power series associated to t and t, respectively. Then y and y are the solutions of the equation
in the ring of formal power series over F2. (See Cristol, Kamae, Mendes-France, and Rauzy 1980.) 2.2.3. To each function /: N -> N with /(0) = 0 and f(n + 1 ) - f(n) positive and odd for n>0, associate the infinite word on A = {a, b] defined by
where vk = iik(b). Each *f is an infinite word without overlapping factor, the mapping /Wa^ is injective and consequently the set of infinite words over A without overlapping factor is not denumerable. (This is a simplified version of a construction of Kakutani, as reported in Gottschalk and Hedlund 1955.)
38
Square-Free Words and Idempotent Semigroups
Section 2.3 2.3.1. For each infinite word a = aoaY — - an- — on A = {a, b), define an infinite word b = bobx • • • bn- • • o n 5 = {a, b, c) by if
anan+x
— aa
if
anan+x
= ab
if
anan+l
= ba
or
bb
If a has no overlapping factor, then b is square-free. If a = t, then b = m (Morse and Hedlund 1944). 2.3.2. Prove Proposition 2.3.2 (See Istrail 1977 and also Dekking 1978.) 2.3.3. Define a sequence wn of words over {a, b, c) by
Then m = lining. (Due to Cousineau unpublished.) *2.3.4. Define over A = {a, b, c) two mappings 77,1 by 7r(a) — abc, ir(b) — bca, 7T(C) — cab and i(d) = ir(d) (dE. A). Extend 77 to A* by *{axa2- -an)
= 7r(a])L(a2)7r(a3)L(a4)"
•.
Then a = ft"(a) is an infinite square-free word. (See Arson 1937; see also Yaglom and Yaglom 1967.) 2.3.5. A morphism a: A* -> B* is called kth power-free if a(w) is kth power-free for each /cth-power-free word w. If a: A* ^> B* is a square-free morphism such that (i) no a(a) is a proper factor of an a(b) (a, bEA), (ii) no a(a) has a nonempty proper left factor that is also a right factor of a(a\ then a is A:th power-free for all k>\. (See Bean, Ehrenfeucht, and McNulty 1979.) 2.3.6. The set of infinite square-free words over three letters is not denumerable (Hint: Use Problem 2.2.3.) 2.3.7. With the notations of Theorem 2.3.1: b is a square-free word such that neither aba nor acbca is a factor of b, if and only if 8(b) has no overlapping factor. (See Thue 1912.)
CHAPTER 3
van der Waerden's Theorem
3.0.
Introduction
This chapter is devoted to a study of van der Waerden's theorem, which is, according to Khinchin, one of the "pearls of number theory." This theorem illustrates a principle of unavoidable regularity: It is impossible to produce long sequences of elements taken from a finite set that do not contain subsequences possessing some regularity, in this instance arithmetic progressions of identical elements. During the last fifty years, van der Waerden's theorem has stimulated a good deal of research on various aspects of the result. Efforts have been made to simplify the proof while at the same time generalizing the theorem, as well as to determine certain numerical constants that occur in the statement of the theorem. This work is of an essentially combinatorial nature. More recently, results from ergodic theory have led to the discovery of new extensions of van der Waerden's theorem, and, as a result, to a topological proof. The plan of the chapter illustrates this diversity of viewpoints. The first section, after a brief historical note, presents several different formulations of van der Waerden's theorem. The second section gives a combinatorial proof of an elegant generalization due to Griinwald. The third section, which concerns "cadences," gives an interpretation of the theorem in terms of the free monoid. In the fourth section is presented a topological proof of van der Waerden's theorem, due to Fiirstenberg and Weiss. The final section is devoted to related results and problems: estimation of various numerical constants, Szemeredi's theorem, conjectures of Erdos, and so on.
3.1.
Classical Formulations
Some forty years after he proved the theorem that bears his name, van der Waerden published an article (1965 and 1971) in which he describes the circumstances of the theorem's discovery. In 1926, in Hamburg, the 39
40
van der Waerden's Theorem 3.1
mathematicians E. Artin, O. Schreier, and B. van der Waerden set to work on the following conjecture of the Dutch mathematician Baudet: PROPOSITION 3.1.1. //1^1 is partitioned into two classes, one of the classes contains arbitrarily long arithmetic progressions.
The conjecture was extended by Artin to the case of a partition of N into k classes. (By a partition of a set E into k classes we mean a family & = {E}9...9Ek} of pairwise disjoint subsets of E whose union is E). The generalization of Baudet's conjecture is thus: PROPOSITION 3.1.2. / / 1^1 is partitioned into k classes, one of the classes contains arbitrarily long arithmetic progressions.
The conjecture was sharpened by Schreier and proved by van der Waerden in the following form: THEOREM 3.1.3. (van der Waerden's theorem). "For all integers there exists an integer N(k,l)EN such that if the set {0, \,...,N(k, /)} is partitioned into k classes, one of the classes contains an arithmetic progression of length /.
It can be shown directly that statements 3.1.2 and 3.1.3 are equivalent. In his account, van der Waerden proposes a diagonal method that amounts to using the following compactness argument. Let us fix the integers k and /. If 3.1.3 is false, then for each nSN there exists a partition &n — {EnX,...,Enk) of {0,...,«} into k classes such that no class of &n contains an arithmetic progression of length /. We associate to each &n a sequence x w £ {0,...,/:}* defined by
if
Now consider the set K = {0,..., A:}N of all sequences with values in (0,..., k) as a topological space with the product topology: K is then a compact metric space that admits the distance function yx(n) — y2(n) for all n such that 0 < n 0. Thus for some n, x „(*) = Xw('o + «) = • • • = X*('o +(* ~ 1)«) =*> consequently the progression {tQ,t0 + a, ...to+{l — \)a) is contained in £„ t, contrary to the hypothesis. So no class of ^contains an arithmetic progression of length /, which contradicts Proposition 3.1.2. Thus 3.1.2 implies 3.1.3. Since 3.1.3 clearly implies 3.1.2 the equivalence of the two statements is proved.
3.2. A Combinatorial Proof of van der Waerden's Theorem The present exposition follows that of Anderson (1976), who proves the following more general result, due to Grunwald (unpublished). 3.2.1. Let S be a finite subset of Nd. For each k-coloring ofNd there exists a positive integer a and a point v inNd such that the set aS + v is monochromatic. Moreover, the number a and the coordinates of the point v are bounded by a function that depends only on S and k {and not on the particular coloring used). THEOREM
(In this statement the word "^-coloring" is synonymous with "partition into k classes." Two points of Nd are of the same color if they belong to the same class of the partition. A monochromatic set is one in which all the elements are of the same color.) Figure 3.1 illustrates the assertion 3.2.1 in the case k — 2 and d — 2. Begin by noting that Theorem 3.2.1 implies van der Waerden's theorem. Indeed,
1 / s
(A
Figure 3.1. Case d = 2, k = 2.
42
van der Waerden's Theorem 3.2
since van der Waerden's theorem is equivalent to statement 3.1.2 it is enough to observe that 3.1.2 follows immediately from 3.2.1 upon taking
Proof (combinatoric) of Theorem 3.2.1. In the following, S denotes a finite subset of Nd and s an element of I\K Kn denotes the cube of side n consisting of those points of Nd all of whose coordinates are less than n. The proof consists of establishing the following statement by induction on 1^1: A(S): for each integer kE N there exists an integer n = n(k) such that for every A:-coloring of Kn9 Kn contains a monochromatic subset of the form aS+v. Theorem 3.2.1 follows from the statements A(S). Indeed, the number a and the coordinates of v can be bounded by n9 since aS + v is contained in Let us begin by observing that A(S) is true if S is empty or consists of a single point, since any set of cardinality < 1 is necessarily monochromatic. Henceforth we will suppose that S is nonempty. To show that A(S) implies A(SU {s}) we introduce an auxiliary statement C(p\ where S is fixed and/?GM. C(p): Let A; E N and sGl\K Then there exists an integer n = n(p,k9s) such that for each A:-coloring of Kn there exist positive integers aQyax,...,ap and a point M E N ^ such that the (p +1) sets a.)
are monochromatic subsets of Kn. These "intermediate" assertions between ^4(5) and A(SU {s}) are proved by induction on p. p = Q: Choose n such that sE.Kn and then set w = (0,...,0) and aQ — \. Then To — {s} is a monochromatic set contained in A^. Passage from p to p + l: Let H = n{p, k,s) be the integer specified in the statement C(p) and let k' — kn . If k colors are available, then there are k' ways to color the cube Kn (since Kn contains nd points). Thus each ^-coloring of Nd induces a A:'-coloring of Nd: Two points u and v will have the same color in the new A;'-coloring if and only if the cubes u + Kn and v + Kn are colored identically in the original A>coloring of ISK It follows from A(S) that there exists an integer n' = n'(k') such that for every A:'-coloring of Kn.y Kn, contains a monochromatic subset of the form a'S + 1 / . Let N—n-\-n\ and consider a A:-coloring of A^y. This can be extended, in arbitrary fashion to a A>coloring of I\IJ, which induces, as described previously, a A:'-coloring of Nd. Since N>n',KN contains a monochromatic subset (with respect to the induced coloring) of the form a'S + v'. This means that the \S\ cubes Kn + a't + 1 / (where / runs over all the points of S) are colored identically with respect to the original A>coloring. However
3.2 A Combinatorial Proof of van der Waerden's Theorem
43
by C(p), Kn contains monochromatic sets
2 a, It follows, upon setting b0 = a' and bt — at_x for K / < ; + l, that the sets
2 are monochromatic. Indeed, if K q < p + 1 , 7^' = T^_ j -I- a'S + 1 / , which is monochromatic by construction. If ^f = 0,
is a singleton and hence monochromatic. Thus C(p + \) holds, with n(p + l9k,s)
= N.
Let us now prove A(SU {s}). Fix the number of colors k and apply C(k): There is an integer n such that for every A:-coloring of Kn, there exist k + 1 monochromatic sets 7J,,..., Tk. By the pigeonhole principle, two of these sets (say, Tr and Tq, where r < q) must be of the same color:
2
af-)s + ( 2fl^j+ f 2 a\
T=u + ( 2 a()s + ( 2 ai Since we supposed £ to be nonempty at the beginning of the induction, S contains at least one point .s0. It follows that the set
2 ^k +f 2
0 0 and S = { 0 , 1 , . . . , n - 1} is called an arithmetic cadence with common difference a. Example. The word 0&btfZ>btf6b 0 there exist integers i<j such that d(Sjx,S'x)<e. We find, upon setting z = S'x and n — j — /, that d(Snz, z) < c, and the result follows. Suppose now that the proposition is true for all positive integers up through/? — 1. We will need two intermediate lemmas. LEMMA 3.4.2. For each e > 0 there exists a finite set of integers such that for all a, bEKyminl^i^Nd(Skia9 b)<e.
ku...,kN
Proof Here is where we use the fact that K is minimal. Recall that the only closed sets—and, by complementation, the only open sets—of K that are stable under S are 0 and K. Thus if co is a nonempty open subset of K, UnGZ5"l(o is a nonempty open subset of K stable under 5, and hence the family (Snco)n(EZ covers K. Since K is compact, a finite subfamily covers K. Let {co!,...,o)n] be a finite covering of K by open sets of diameter <e. Then, for I = 1 , . . . , H , there exists a finite family {Sn'Jo)i}]^j^ri that covers {Snijo)i}l^J^r. K. Let a.bEK. Then bGo)t for some i"E{l,...,/i}. Since l/ n covers K, aES" >co/ for someyEfl,...,^-}. Therefore S~ ' ^Eco, and thus
d(S~n'Ja, b)<e. It follows that min
min d(S~n'
ja,b)<e,
which proves the lemma. LEMMA
such that Proof
•
3.4.3. For all e> 0 and for all a^K there exist bGK and n>0 d(Snb,a)<e,...,d(Spnb,a)<e.
B y L e m m a 3.4.2 there exist integers kx,...,kN k
such that for all
3.4 A Topological Proof of van der Waerden's Theorem
47
Since each Ski is uniformly continuous on K, there exists a positive real number i\ such that d{a,a') 2 fixed, these upper bounds are not even primitive recursive functions of / ) . Moreover, no "reasonable" upper bounds are known. The first few values of N(k,l) are known:
1 2 3 4 5
1
2
3
4
5
1 2 3 4 5
1 3 9 35 178
1 4 27
1 5
1 6
Because upper bounds appear to be out of reach, most of the work has been aimed at finding lower bounds for N(k, I). In 1952 Erdos and Rado proved the following result: N(k,l)>(2(l-l)k'-l)X/2
(3.5.1)
A combinatorial proof of (3.5.1) will be given shortly. This bound was improved in 1960 by Moser, who used a constructive method. N(k,l)>lkclo*k,
(3.5.2)
where c is a constant. The other known results concern N(291). In 1962 Schmidt used a probabilistic method to prove N(2J)>2l-«n(W]/\
(3.5.3)
where c is a constant. The best lower bound known was found by Berlekamp (1968), by means of a constructive method using the finite fields GF(2l): If I is a prime andl>5,
then N(29 / + 1 ) > I2l.
(3.5.4)
In one of his survey papers, Erdos (1977) also mentions the following lower bound, which is valid for every /. N(29I)>c2'9
(3.5.5)
where c is a constant. It appears that these bounds are still quite rough. Indeed, Erdos has conjectured that N(2, l)x/l -^ oo, but, so far as we know, this has not been proved.
50
van der Waerden's Theorem 3.5
To prove the lower bound, Eq. (3.5.1): Set N = N(k91 +1). We will begin by counting the arithmetic progressions of length /-hi in {1, . . . , # } . First of all, the common difference r of such a progression can vary between 1 and [(AT — 1)//J. Second, the first term of the progression lies between 1 and TV — rl. It follows that the number of progressions is
-=
2 '\~r\
and thus
N2-\
N2
On the other hand, the number of partitions of {1,...,7V} into k classes is kN. Let K ax < a2- • < a/+, ^ N be an arithmetic progression with (/ +1) terms. The number of partitions of {1,... ,N) into k classes such that one of the classes contains this progression is kNl (indeed, the class of each element of {\,...9N}\{a29...,al+]} can be chosen arbitrarily, then the elements fl2,...,a/+1 must go into the class containing ax). It follows that the number of partitions of {1,...,7V} into k classes such that at least one of the classes contains an arithmetic progression of length /-hi is at most MkN~l. However, by the definition of N every partition of {1,...,JV} into k classes satisfies this property. Thus kN<MkN~l, from
which it follows that kl^M
(2lk l ) l/2 .
More than forty years ago, Erdos and Turan (1936) introduced another numerical constant connected with van der Waerden's theorem: Let us denote by r,(«) the smallest integer such that every subset of {1,...,«} with r/(n) elements contains an arithmetic progression of length /. For example the reader can check that r4(10) = 9 as {1,2,3,5,6,8,9,10} contains no arithmetic progression of length 4, but every subset of {1,..., 10} with nine elements contains such a progression. This example is taken from an article by Wagstaff (1979) that contains a table of the first few values of rt{n)— 1. As Szemeredi (1975) has pointed out, there is good reason to study these numbers. Estimation of r,{n) is in itself interesting, and moreover, it can eventually lead to upper bounds for N(k, /). Indeed, it is easy to see that if rj{n)/n (m) + r,(w),
(3.5.6)
proved by Erdos and Turan in 1936, leads to a fairly easy proof (see Problem 3.5.2) that
inf^i = c/ w>0
(3.5.7)
n
(cf. Behrend 1938). Erdos and Turan (1936) showed that c3 0. Behrend's result was generalized by Rankin (1960), who found the following lower bounds
f^j
(3.5.11)
The best upper bound for r3(n) now known is due to Roth (1952): r 3 (w)< f °" . 3V ' loglog/z
(3.5.12) '
v
This inequality implies, of course, that c3 = 0. The equality c4 = 0
(3.5.13)
was proved for the first time by Szemeredi (1969) by purely combinatoric methods employing van der Waerden's theorem. A bit later Roth (1970, 1972) gave an analytic proof which did not use van der Waerden's theorem.
52
van der Waerden's Theorem 3.5
Furthermore, Szemeredi (1975) said that Roth's method probably gives an upper bound of the following form: 4
(
) , log{k)n
where k is a sufficiently large (but fixed) integer and where log(/c)« designates the A;-fold iterated logarithm. Finally, Szemeredi (1975) proved the conjecture of Erdos and Turan: c, = 0
for all/
(3.5.14)
and collected the $1,000 reward offered by Erdos. Recently, Furstenberg (1977), using ergodic theory, gave another proof of this result. (In fact Furstenberg proves a much more general result.) Nevertheless, the problem of precise estimation of rt(n) remains open. In a more direct formulation of Szemeredi's theorem we say that a subset S of fol has density d if hm
- — d.
n-+oo
n
Szemeredi's theorem can be stated as follows: / / 5 is a subset of N with density > 0, then S contains arbitrarily long arithmetic progressions. Erdos has offered $3,000 for the resolution of the following conjecture, which generalizes the preceding statement: If S is a subset ofN\{0} such that
s<ES *
then S contains arbitrarily long arithmetic progressions. This conjecture, if true, would prove the conjecture on arithmetic progressions of prime numbers (take S to be the set of primes). Let us point out, with regard to prime numbers, that there exist infinitely many arithmetic progressions consisting of three primes (Chowla 1944), and that there exists an arithmetic progression consisting of seventeen primes (Weintraub 1977). However, little is known on Erdos's conjecture; see Gerver (1977).
Notes Van der Waerden's original proof can be found in van der Waerden (1927). See also van der Waerden's personal account (1965, 1971). An exposition by Khinchin (1952) and a short proof by Witt (1951), Graham
Problems
53
and Rothschild (1974), Deuber (1982) are also available. The generalization by Anderson (1976) given in the text follows essentially the arguments of Witt and Graham-Rothschild. Erdos's successive survey papers (1963, 1965, 1974, 1977, 1979)—with Spencer in 1974 and with Graham in 1979—give a good idea of the advances on van der Waerden's theorem and related topics. In particular the reader is referred to the paper by Erdos and Graham (1979) for a number of interesting questions not discussed here and for further references. See also the recent book by Graham, Rothschild and Spencer (1980). The ergodic proof of Szemeredi's theorem is due to Furstenberg (1977). See also Thouvenot (1978). An analogue of Szemeredi's theorem in higher dimensions has been proved by Furstenberg and Katznelson (1978). The topological proof given in the text follows Furstenberg and Weiss (1978), where other extensions are discussed. Girard (1982) has shown that, after a slight modification, the topological proof leads to an upper bound for N(k,l). For an application of van der Waerden's theorem to number theory, see Shapiro and Spencer (1972).
Problems Section 3.1 3.1.1. Let a = (ai)isN be a strictly increasing sequence of positive integers such that there exists an integer M with ai+} — a{ < M for all /E fol. Show (without Szemeredi's theorem) that a contains arbitrarily long arithmetic progressions. {Hint. Consider the classes
for 0 < k < M - 1 and observe that Q C Q + k.) (See Brown 1969, Rabung 1970.) 3.1.2. Show that if fol is partitioned into two classes, either one class contains arbitrarily long sequences of consecutive integers or both classes contain arbitrarily long arithmetic progression. (Observe that if the first condition does not hold, then there exists an integer M such that every interval of length M meets both classes. Now apply Problem 3.1.1.) Section 3.3 3.3.1. Let u be an infinite word and T a subset of N\{0}. We say that T is a cadence for u if all the letters whose position in u belong to T are identical. The definitions of cadence of type S and arithmetic cadence are the same as in the finite case. Show that the infinite word aba2b2a3b3- -anbn--- contains no arithmetic cadence of infinite order.
54
Problems
3.3.2. Let u be an infinite word whose zth letter is a or b, depending on whether the first occurrence of 1 (reading from right to left) in the binary representation of i! is an even or odd position. (Formally a b
if if
i! = n2 2/ i! = n2 2 ' + 1
with n odd with n odd'
Show that for all d>0 there exists an integer n(d) such that u contains no arithmetic cadence with common difference d and order ^n(d). (Justin, unpublished; see another method in T. C. Brown 1981.) Section 3.5 3.5.1. Show that if N is partitioned into k classes, one of the classes contains arbitrarily long geometric progressions (Hint: Given a partition & = {El9...9Ek] of M, consider the partition &'={E[9...,E'k} defined by n E E[ if and only if 2 n E Et. Then apply van der Waerden's theorem.) 3.5.2. Show that if a sequence (un)n^0 of positive real numbers satisfies the triangle inequality ur+s ^ur + us for all r, s > 0, then lim n ^^u n /n — win^oun/n. (Hint: Show that for n = noq + r where
CHAPTER 4
Repetitive Mappings and Morphisms
4.0.
Introduction
This chapter is devoted to the study of a special type of unavoidable regularities. We consider a mapping E from A+ to a set £ , and we search in a word w for factors of the type wxw2- • • wn9 with
