Infinite words containing squares at every position

Infinite w ords con taining squares at ev ery p osition James Curri e ∗ and Narad Ramp ersad † Department of Mathemati cs and Statisti cs Universit y of Winnip eg 515 P or t age Aven ue Winnip eg , Ma nitoba R3B 2E9 (C anada) j.cu rrie @uwin nipeg.ca n.ra mper sad@u winnipeg.ca August 13, 2021 Abstract Ric homme a sked the follo wing question: what is the infi m um of the real n umbers α > 2 suc h that there exists an infinite wo rd that a voids α -p ow ers but con tains arb i- trarily large squares b eginning at ev ery p ositio n? W e r esolv e this question in the case of a binary alph ab et b y sho wing that the answ er is α = 7 / 3. 1 In tro duction W e consider the following question of Ric homme [17]: what is t he infimum of the real num bers α > 2 such that there exists an infinite w ord that a v oids α -p o w ers but con tains a r bit r a rily large squares b eginning at ev ery p o sition? As w e shall see, ov er the binary alphab et, the answ er to Ric homme’s ques tion is α = 7 / 3. First w e recall some basic definitions. If α is a rational num ber, a w ord w is an α - p ower if there exist w ords x and x ′ , with x ′ a prefix of x , suc h that w = x n x ′ and α = n + | x ′ | / | x | . W e refer to | x | as a p erio d of w . An α + - p ower is a w ord that is a β - p o wer for some β > α . A w ord is α - p ower-fr e e (resp. α + - p ow er-fr e e ) if none of it s sub w ords is an α -p o w er (resp. α + -p ow er). A 2-p o wer is called a squar e ; a 2 + -p ow er is called an overlap . The motiv ation for Richomme’s question comes from the observ at ion that there exist ape- rio dic infinite binary w ords that con tain arbitrarily larg e squares starting at ev ery p o sition. F or instance, all Sturmian w ords hav e this prop e rty [1, Proposition 2]. ∗ The author is s uppo r ted b y an NSER C Disco very Grant. † The author is s uppo r ted b y an NSER C P os tdo c to ral F ellowship. 1 Certain Sturmian w ords are not only ap erio dic, but also av o id α -p o we rs fo r some real n um b er α . F or instance, it is w ell-know n that the F ib onacci w ord f = 0100101001001 0 1001010 · · · con tains no (2 + ϕ ) - p ow ers [14] (see a lso [13]), where ϕ = (1 + √ 5) / 2 is the golden ratio. By con trast, the Th ue–Morse w ord t = 01101001 1 00101101 0010110 · · · is ov erlap-free [20] but do es not con tain squares b eginning at every p osition (It is an easy exercise to sho w that t do es not b egin with a square). The squares o ccurring in the Th ue– Morse w ord hav e b een c har a cterized b y P ansiot [15] and Brlek [4], and the p ositions at whic h they o ccur w ere studied by Brown et al. [5]. Saari [18 ] also studied infinite w or ds containing squares (not necessarily arbitrarily large) b eginning at e very p osition. He calls suc h w ords squar eful (though he imp oses the additional condition that the w ord contain o nly finitely ma ny distinct minimal squares). 2 Ov e rlap-free and 7 / 3 -p o w er-free squares W e b egin b y reviewing what is known concerning the o v erlap-fr ee binary squares. Subse- quen tly , w e shall generalize this c haracterization to the 7 / 3 -p ow er-free binary squares. Let µ denote the Thue–Mors e mo r phism: i.e., the mor phism that maps 0 → 0 1 and 1 → 10. Define sets A = { 00 , 11 , 01001 0 , 101 101 } and A = [ k ≥ 0 µ k ( A ) . The set A is the set of squares app earing in the Th ue–Morse w ord. Shelton and Soni [19] c haracterized the ov erlap-free squares (the result is also attributed to Th ue b y Berstel [2 ]), as being the conjugates of the w ords in A (A conjugate of x is a w ord y suc h that x = uv and y = v u for some u, v ). An immediate application is the following t heorem: Theorem 1. I f w is an infinite overlap-fr e e bin ary wor d, then ther e is a p osition i such that w do es not c o n tain a squar e b e g i nning at p osition i . Pr o of. An easy computer searc h suffices to v erify that any ov erlap-free w o r d o v er { 0 , 1 } of length greater than 36 mus t con tain the sub w ord 0 10011. Let i denote a ny p osition at w hich 010011 o ccurs in w . W e claim that no square b egins at p osition i . Supp ose to the con trary that xx is suc h a square. By Shelton and Soni’s result, except for the squares that are conjugates of A , eve ry o v erlap-free sq uare xx has | x | ev en. It follows that xx is of the form 0 y 10 y 1, where y ∈ { 01 , 10 } ℓ for some ℓ . Ho w ev er, this forces xx to b e follow ed b y 0 in w , so that w e hav e the ov erlap xx 0 as a sub w ord of w , a con tra diction. 2 The next theorem generalizes the c haracterization of Shelton and Soni. Theorem 2. The 7 / 3 -p ower-f r e e binary squar es ar e the c onjugates of the wor ds in A . W e defer the pro of of Theorem 2 to Section 4. How ev er, w e end this section b y pro ving an analogue, for 7 / 3-p ow er-free w ords, of a w ell-known “progression lemma” for ov erlap- free w ords [3, 7, 10, 11, 16, 19] that w e shall need later. The w ords µ n (0) and µ n (1), n ≥ 0, are kno wn as Morse blo cks . Note that the rev erse o f a Morse blo c k is a Morse blo ck . Lemma 3. L et w = uv xy b e a binary 7 / 3 -p o w er-fr e e wor d with | u | = | v | = | x | = | y | = 2 n . If u and v ar e Morse bl o cks, then x is a Morse blo ck. Pr o of. The pro of is by induction on n . Clearly , the result ho lds for n = 0. W e hav e either (1) w = u ′ u ′′ u ′ u ′′ pq r s or (2) w = u ′ u ′′ u ′′ u ′ pq r s , where u ′ and u ′′ are distinct Morse blo c ks of length 2 n − 1 and | p | = | q | = | r | = | s | = 2 n − 1 . By induction, p , q , and r are also Morse blo c ks. W e m ust sho w that p 6 = q . In case (1), pq = u ′ u ′ creates the 5 / 2 - p ow er u ′ u ′′ u ′ u ′′ u ′ , and pq = u ′′ u ′′ creates the cub e u ′′ u ′′ u ′′ . In case ( 2 ), pq = u ′ u ′ creates the cub e u ′ u ′ u ′ ; and if pq = u ′′ u ′′ , then r = u ′ creates the 7 / 3-p ow er u ′ u ′′ u ′′ u ′ u ′′ u ′′ u ′ , and r = u ′′ creates the cub e u ′′ u ′′ u ′′ . Th us, p 6 = q , as required. 3 Infinite w ords con tain ing squ ares at ev ery p osition W e b egin b y sho wing that the answ er to R ic homme’s question is at most 7 / 3. Theorem 4. T h er e exists an infinite (7 / 3) + -p ower- f r e e binary wor d that c ontains arbitr arily lar ge squar es b e ginn i n g at every p o s i tion . Pr o of. W e rely on the exis tence of a (7 / 3) + -p ow er-free morphism; that is, a morphism f suc h that f ( w ) is (7 / 3) + -p ow er-free whenev er w is (7 / 3) + -p ow er-free. K olpak ov , Kuchero v, and T arannik ov [12] hav e giv en an example of suc h a mor phism f : 0 → 01101001 10010011 01001 1 → 10010110 01001100 10110 . F or con v enience w e w ork instead with the following morphism g : 0 → 01101001 10110011 01001 1 → 10010110 01101100 10110 , whic h we obtain by setting g (0) = f (1) and g (1) = f (0), where the o v erline denotes binary complemen ta tion. W e no w define a sequence of w o r ds ( A n ) n ≥ 0 as follows. Let A 0 = 011011 0. F or n ≥ 0 define A n +1 = (011010) − 1 g ( A n ), where the notatio n (011010) − 1 x denotes the w ord obtained 3 b y remo ving t he prefix 011010 fro m t he w ord x . Since g ( A n ) alwa ys b egins with 011010 , this op eration is w ell-defined. The sequence ( A n ) n ≥ 0 th us b egins A 0 = 0 1 10110 A 1 = 0 1 10110011 01001100 10110011 0 1100101101001011001101100 1 0110 · · · . . . Observ e that since g is (7 / 3) + -p ow er-free, A n is a lso (7 / 3) + -p ow er-free. W e first show that as n → ∞ , A n con v erges to an infinite limit word w ; second, w e show that w con tains arbitrarily long squares b eginning at ev ery position. W e sho w by induc tio n on n that A n is a prefix of A n +1 . Clearly this is true for n = 0. Recall that A n +1 = (011010) − 1 g ( A n ). Inductiv ely , A n − 1 is a prefix of A n , so g ( A n − 1 ) is a prefix of g ( A n ). Th us, (011010) − 1 g ( A n − 1 ) = A n is a prefix of (011010) − 1 g ( A n ) = A n +1 , as required. W e conclude that A n tends, in the limit, to a (7 / 3) + -p ow er-free w ord w . T o see that w con tains arbitra rily long squares b eginning at ev ery p osition, first let u = 011010 and observ e that for ev ery n ≥ 1 w e ha ve A n = u − 1 g ( u ) − 1 [ g 2 ( u )] − 1 · · · [ g n − 1 ( u )] − 1 g n (0110110 ) . Let v = g n − 1 ( u ) g n − 2 ( u ) · · · g ( u ) u, so that A n = v − 1 g n (0110110 ) , and observ e that | v | = 6 n − 1 X j =0 21 j = 6  21 n − 1 21 − 1  < 21 n = | g n (0) | . W e see then that v is a prefix of g n (0). W rite g n (0) = v x , so that A n = x g n (11) v xg n (11) v x . It follo ws that fo r 0 ≤ j < | x | = 21 n − (3 / 10) (21 n − 1), A n , and hence w , con tains a square of length 6 · 21 n b eginning at position j . Since n ma y b e tak en to b e arbitrarily large, the result follow s. The result of the previous theorem is optimal, as w e no w demonstrate. Theorem 5. If w is an infinite 7 / 3 - p ower-fr e e binary w o r d, then ther e is a p osition i s uch that w do es n ot c ontain arbitr a rily la r ge squar es b e ginnin g at p osi tion i . Pr o of. As in the pro of of Theorem 1, a n easy computer searc h suffices to v erify that an y 7 / 3-p ow er- free w ord ov er { 0 , 1 } of length greater than 39 m ust con tain the sub word 010011 . Let i b e a p osition at whic h there is an o ccurrence of 0 10011 in w . Supp ose that there is a square xx b eginning at p osition i . By Theorem 2, xx is a conjugate of a w ord in A . In particular, xx / ∈ A ; that is, xx is a non-identity c onjugate of a w ord in A . Case 1: xx is a conjugate of either µ k (00) or µ k (11) for some k . Then xx is a conjug ate of a w ord of the form uv uv , where u and v are Morse blo c ks of the same length. Witho ut loss of generalit y , w e write xx = u ′′ v uv u ′ , where u ′ u ′′ = u and u ′ 6 = ǫ 6 = u ′′ . 4 Supp ose that y y is ano t her square b eginning at p osition i . Supp ose further that there are arbitrarily large squares b eginning a t p osition i , so that we may c ho ose | y | > | xx | . W e see then that there is an o ccurrence of u ′′ v uv u ′ at p osition i + | y | . Considering this later o ccurrence o f u ′′ v uv u ′ , and observing that Morse blo c ks of a giv en length are uniquely iden tified by their first letter (as w ell as b y their la st letter), w e ma y apply Lemma 3 to this later o ccurrence of u ′′ v uv u ′ to conclud e that the v uv of this occurrence is b oth prece ded a nd follo we d b y the Morse blo c k u . Thus , w con tains t he 5 / 2-p ow er uv uv u , a contradiction. Case 2: xx is a conjugate o f either µ k (010010) or µ k (101101) for some k . By a similar argumen t as in Case 1, w e may supp ose that xx has one o f the forms u ′′ v uu v u u ′ , u ′′ v v uv v u ′ , or u ′′ uv u uv u ′ , w here u and v ar e Morse blo c ks of the same length, u ′ u ′′ = u , and u ′ 6 = ǫ 6 = u ′′ . As before, w e suppose the existenc e of a square y y b eginning at p osition i , where | y | > | xx | . Then there is a later o ccurrence of xx at p osition i + | y | . Applying Lemma 3 to this later o ccurrence of xx , we deduce the existenc e of one of the 7 / 3-p ow ers uv u uv uu , uv v uv v u , or uuv uuv u , a con tra diction. All cases yield a con tradiction; w e conclude that there do es not exist arbitrarily large squares b eginning at p osition i , as required. It is p ossible, ho w ev er, to ha ve an infinite 7 / 3-p ow er-f ree binary w ord with squares b egin- ning at ev ery p osition; w e are only preve nted fro m ha ving a r bit r a rily large s quares beginning at ev ery p osition. Theorem 6. Ther e exists a n infinite 7 / 3 -p ower- f r e e binary wor d that c on tains squar es b e- ginning at every p osition. Pr o of. W e show that a word constructed b y Currie, Ramp ersad, and Shallit [6] has the desired pro p ert y . The construction is as follow s. W e define the fo llowing sequence of words : A 0 = 00 and A n +1 = 0 µ 2 ( A n ), n ≥ 0. The first few terms in this sequ ence are A 0 = 00 A 1 = 00110011 0 A 2 = 00110011 0 10011001 01100110 100110010110 . . . Currie et al. sho wed that as n → ∞ , this se quence con v erges to an infinite word a , and further, a is 7 / 3-p ow er-f ree. W e sho w that a con tains squares b eginning at ev ery po sition. W e claim that for n ≥ 0, a con tains a word of the fo rm xxx ′ at p osition (4 n − 1) / 3, where | x | = 4 n +1 and x ′ is a prefix of x of length 4 n . Observ e that f o r n ≥ 0, b y the construction of A n +1 , w e hav e A n +1 = 0 µ 2 (0) µ 4 (0) · · · µ 2( n − 1) (0) µ 2 n ( A 1 ) . Ho w ev er, µ 2 n ( A 1 ) = xxx ′ , where x = µ 2 n (0011) has length 4 n +1 ; x ′ = µ 2 n (0) is a prefix of x of length 4 n ; and µ 2 n ( A 1 ) o ccurs at p osition n − 1 X i =0 4 i = 4 n − 1 3 , 5 as claimed. It follow s that for i ∈ [(4 n − 1 ) / 3 , (4 n +1 − 1 ) / 3 − 1], a contains a square of length 2 · 4 n +1 at p osition i . This completes the pro o f. Although the w ord constructed in the pro of of Theorem 6 contains squares b eginning at ev ery p osition, it is not squar eful in the sense of Saari [18], since it do es not con tain only finitely man y minimal squares . F or a squareful w ord, there exists a constan t C suc h that at ev ery p osition there is a square of length at most C . T o see that this do es not hold fo r the w ord a constructed ab o ve , w e note that b y the factorization theorem of Karh um¨ aki a nd Shallit [9] (Theorem 8 b elo w), an y infinite 7 / 3- p ow er-free binary w ord con tains o cc urrences of µ n (0) for arbitrarily large n . Ho w ev er, µ n (0) is a prefix of the Th ue–Morse w or d, and we ha v e already noted in the in tro duction that the Th ue–Morse w ord do es not b egin with a square. Th us there cannot exist a constant C b ounding the length of a minimal square in a , so a is no t squareful. In general, no infinite 7 / 3-p ow er-free binary w ord can b e squareful. The result of T heorem 6 can b e s trengthened by applying a mor e general construction of Currie et al. [6]. Theorem 7. F or every r e al numb er α > 2 , ther e exists an infinite α -p owe r-fr e e bina ry wor d that c ontains squar es b e ginnin g at every p o s i tion . Pr o of. Since Theorem 6 establishe s the result for α ≥ 7 / 3, w e only consider α < 7 / 3. W e recall the follo wing construction of Currie et al. [6, Theorem 14]. Let s ≥ 3 and t ≥ 5 be in tegers suc h that 2 < 3 − t/ 2 s < α , and suc h that the w ord obtained b y removing the prefix of length t from µ s (0) b egins with 00. Let β = 3 − t/ 2 s . W e construct sequences of w or ds A n , B n and C n . Define C 0 = 00 and let u b e the prefix of length t of µ s (0). F or each n ≥ 0: 1. Let A n = 0 C n . 2. Let B n = µ s ( A n ). 3. Let C n +1 = u − 1 B n . Currie et al. sho w ed t ha t the C n ’s conv erge t o an infinite w ord w that is β + -p ow er-free, and hence, α -p o wer-free. F or n ≥ 1, w b egins with a prefix of the form u − 1 µ s (0)[ µ s ( u )] − 1 µ 2 s (0) · · · [ µ ns ( u )] − 1 µ ( n +1) s (0) µ ( n +1) s (00) . Let u ′ = u − 1 µ s (0). Th us, for n ≥ 1 , w con tains the word µ ns ( u ′ ) µ ( n +1) s (00) at p osition F n = 2 s n − 1 X i =0 2 is − t n − 1 X i =0 2 is = (2 s − t )  2 ns − 1 2 s − 1  , where µ ns ( u ′ ) is a suffix of µ ( n +1) s (0). Letting | u ′ | = t ′ and defining G n = | µ ns ( u ′ ) | = t ′ · 2 ns , w e hav e F n = t ′  2 ns − 1 2 s − 1  < G n . 6 Since u ′ is a suffix o f µ s (0), and since w b egins with u ′ µ s (00), w e see that for j ∈ [0 , t ′ − 1], ev ery subw ord of w of length 2 s +1 starting at p osition j is a square. Similarly , for n ≥ 1, w e see that t here is a square of length 2 ( n +1) s +1 starting at p osition j for every j ∈ [ F n , F n + G n − 1]. Since F n < G n , there is th us a square at ev ery p osition o f w . 4 Pro o f of Theo r e m 2 In this section we giv e the pro of of Theorem 2 . W e b egin with some lemmas, but first w e recall the factorization theorem of Karh um¨ aki and Shallit [9]. Theorem 8 (Karh um¨ aki a nd Shallit ) . L et x ∈ { 0 , 1 } ∗ b e α -p ower-fr e e, 2 < α ≤ 7 / 3 . Then ther e exist u, v ∈ { ǫ, 0 , 1 , 00 , 11 } and an α -p ower-fr e e y ∈ { 0 , 1 } ∗ such that x = uµ ( y ) v . Lemma 9. L et xx ∈ { 0 , 1 } ∗ b e 7 / 3 -p ower-fr e e. If xx = µ ( y ) , then | y | is ev en. Conse quently, y is a squar e. Pr o of. Supp ose to the contrary that | y | = | x | is o dd. By an exhaustiv e en umeration one v erifies that | x | ≥ 5. But then xx con tains t w o o ccurrences of 00 (or 11) in p o sitions of differen t parities, whic h is imp ossible. The next lemma is a v ersion of Theorem 8 sp ecifically applicable to squares. Lemma 10. L et xx ∈ { 0 , 1 } ∗ b e 7 / 3 -p ower-fr e e. If | xx | > 8 , then e i ther (a) xx = µ ( y ) , wher e y ∈ { 0 , 1 } ∗ ; or (b) xx = aµ ( y ) a , wher e a ∈ { 0 , 1 } an d y ∈ { 0 , 1 } ∗ . Pr o of. Applying Theorem 8, w e write xx = uµ ( y ) v . W e first sho w tha t | u | = | v | ≤ 1. Supp ose that u = 00. Then xx b egins with one of the words 000, 00100, or 0010 10. The first and third w ords contain a 7 / 3-p ow er, a con tra diction. The s econd w ord, 00100, cannot o ccur later in xx , as that w ould also imply the existence of a 7 / 3- p o wer. W e conclude u 6 = 00, and similarly , u 6 = 11. A similar argumen t also holds for v . Since | xx | is ev en, w e must therefore ha v e | u | = | v | , as required. If | u | = | v | = 0, then we hav e established (a ). If | u | = | v | = 1 , it remains to sho w that u = v . If u = v , then w e hav e xx = uµ ( y ) u . Since x b egins and ends with u , w e ha v e x = uµ ( y ′ ) = µ ( y ′′ ) u , where y = y ′ y ′′ . Let z ′ = µ ( y ′ ) and z ′′ = µ ( y ′′ ). Then z ′′ b egins w ith u and hence with u u . This implies that z ′ b egins with u and hence with uu . This in turn implies that z ′′ b egins with uuuu , and hence that z ′ b egins with uuuu . No w w e see t hat x begins with the 5 / 2-p o wer u uuuu , whic h is a con t r adiction. W e conclude that u 6 = v , a s required. W e are no w ready to prov e Theorem 2. 7 Pr o of of The or em 2. Let xx b e a minimal 7 / 3-p o wer-free square that is not a conjugate of a w ord in A . That | xx | > 8 is easily ve rified computationally . Applying Lemma 10 leads to t w o cases. Case 1: xx = µ ( y ). By Lemma 9, y is a square. F urthermore, y is not a conjugate of a w ord in A , con tradicting the minimality of xx . Case 2: xx = aµ ( y ) a . Then aa µ ( y ) = µ ( ay ) is also a square z z . W e show that z z is 7 / 3- p o wer-free, and consequen tly , b y Lemma 9 , that ay is a 7 / 3-p o w er-f ree square, con tradicting the minimalit y of xx . Supp ose to the con trary that z z con tains a 7 / 3-p ow er s = r r r ′ , where r ′ is a prefix o f r and | r ′ | / | r | ≥ 1 / 3. The w ord s m ust o ccur at t he b eginning of z z and we m ust ha v e | s | > | x | ; otherwise, xx w ould con tain an o cc urrence of s , con tradicting t he assumption that xx is 7 / 3-p ow er-free. W e ha ve four cases , dep ending on the relative sizes of | r | and | z | , as illustrated in Figures 1 – 4. By analyzing the ov erlaps b etw een z z and r r r ′ , denoted X in the figures, w e deriv e a con tradiction in each case. Case 2a: 2 | r | < | z | ≤ 2 | r | + | r ′ | / 2 (Figure 1). In this case , r ′ has a prefix X that is also a suffix of z . Then X is also a prefix of z and X X is a prefix of r ′ . Consequen tly , X X is a prefix of z and xx con tains the cub e X X X . This is a con tradiction. Case 2b: | z | > 2 | r | + | r ′ | / 2 (Figure 2). In this case, r ′ is of the form X Y X , where X Y is a suffix of z and X is a prefix of z . Since r ′ = X Y X , then z has X Y X as a prefix. In particular, xx con tains a 7 / 3- p ow er-free square X Y X Y . By the assumed minimalit y of xx , X Y X Y is a conjugate of a w ord in A . If X Y X Y = µ k ( w ), where w is a conjugate of a w o rd in A (not A !) , then X Y X Y can b e written either as A 1 A 1 or as A 1 A 2 A 3 A 1 A 2 A 3 , where the A i ’s are a ll Morse blo ck s of the same length. Since X Y X Y is follo w ed b y X in z z , and since the Morse blo c ks of a given length a re uniquely iden tified b y their first letter, b y Lemm a 3 X Y X Y is follo w ed b y the Morse blo ck A 1 , creating either a cub e or a 7 / 3 -p ow er in xx , con trary to our assumption. If X Y X Y 6 = µ k ( w ), where w is a conjugate of a word in A , then w e ma y write X Y X Y as one of uB AB v , uB AAB Av , uB B AB B v , or uAB AAB v , where A and B are Morse blo c ks, B = A , and v u = A . If X Y X Y = uB AB v , then, since u is a non-empty suffix of A and v is a non-empt y prefix of A , and since the Morse blocks o f a given le ngt h a re unique ly iden tified by their first letter (as w ell a s b y their la st letter), w e can a pply Lemma 3 to conclude that B AB is preceded and follow ed b y A . Th us xx con tains the 5 / 2-p o wer AB AB A , con t r a ry to our assumption. Similarly , if X Y X Y = uB AAB Av , then B AAB A is preceded and follo w ed b y A , creating the 7 / 3-p ow er AB AAB AA . The other p ossibilities for X Y X Y lead to the existence of a 7 / 3-p ow er in xx by a similar argument. Case 2c: | z | ≤ 3 / 2 · | r | (Figure 3). In this case, r has a prefix X t hat is also a s uffix of z . Then X is also a prefix of z and X X is a prefix of r . Consequen tly , X X is a prefix of z and xx con tains the cub e X X X . This is a con tradiction. Case 2d: 3 / 2 · | r | < | z | < 2 | r | (Figure 4). In this case, r has a suffix X that is also a prefix of z . Then X is also a prefix o f r and r ′ b egins with some prefix X ′ of X . Consequen tly , 8 X r r r’ X X X X X X X X z z Figure 1: The case where 2 | r | < | z | ≤ 2 | r | + | r ′ | / 2 X z z X X Y Y X Y X r r r’ X X X X Y Y Figure 2: The case where | z | > 2 | r | + | r ′ | / 2 X X z z r’ r r X X X X X X X Figure 3: The case where | z | ≤ 3 / 2 · | r | X X z z X X X X X’ X’ X X r r r’ Figure 4: The case where 3 / 2 · | r | < | z | < 2 | r | 9 r r , and indeed xx , con t a ins a sub w ord X X X ′ that is at least a 7 / 3- p o we r. This is a con tradiction. Since in all cases w e hav e deriv ed a con tradiction b y show ing that xx contains a 7 / 3- p o wer, w e conclude that our assumption t ha t z z contains a 7 / 3 -p ow er is false. Recalling that z z = µ ( ay ) a nd that ay is necessarily a square, w e conclude that ay is a 7 / 3-p ow er-free square, con tradicting the minimalit y of xx . W e conclude that there exists no 7 / 3-p ow er-fr ee square xx that is not a conjuga te of a word in A . W e claim that the constan t 7 / 3 in Theorem 2 is b est p ossible. T o see this, note that the w ord 01101001 10110010 11001101 0 0110110010110 is a ( 7 / 3) + -p ow er-free square, but is not a conjugate of a w ord in A . 5 Conclus ion W e ha ve only considered w ords o v er a binary alphab et. It remains to consider whether similar results hold o v er a larger alphab et. F o r ins tance, do es there exist an infinite o v erlap- free ternary word that con tains squares b eginning at ev ery p osition? Ric homme observ es that o ver an y alphab et there cannot exist an infinite o verlap-free w ord containing infinitely man y squares at eve ry p osition. He points out that this follo ws from the follo wing res ult of Ilie [8, Lemma 2]: In an y word, if v v and u u ar e tw o squares at p osition i and w w is a square at p osition i + 1, then either | w | = | u | or | w | = | v | or | w | ≥ 2 | v | . An easy conseq uence of this result is that in an y infinite word, if infinitely man y distinct squares b egin at p o sition i and w w is a square b eginning at p o sition i + 1, then | w | = | u | for some square uu o ccurring at p osition i , and hence there is an o v erlap a t p o sition i . Ac kno wled gmen ts The authors wish t o thank Gw ´ ena ¨ el Ric homme for suggesting the problem. W e also thank him for reading an earlier draft of this pap er and pro viding man y helpful commen ts and suggestions. References [1] J.- P . Allouc he, J. L. Da vison, M. Queff ´ elec, L. Q. Zam b o ni, “T ranscendence of Sturmian or morphic con tin ued fractions”, J. Numb er The ory 91 ( 2 001), 3 9–66. [2] J. 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