On the critical exponent of generalized Thue-Morse words

For certain generalized Thue-Morse words t, we compute the"critical exponent", i.e., the supremum of the set of rational numbers that are exponents of powers in t, and determine exactly the occurrences of powers realizing it.


Introduction
It is a well-known fact that the Norwegian mathematician Axel Thue  was the first to explicitly construct and study the combinatorial properties of an infinite overlap-free word over a 2-letter alphabet, obtained as the fixpoint of the morphism µ : {a, b} * → {a, b} * defined by µ(a) = ab; µ(b) = ba: For a modern account of his papers, see Berstel [3]. Rediscovered by M. Morse in 1921 in the study of symbolic dynamics, this overlap-free word is now called the Thue-Morse word. This "ubiquitous" sequence, already implicit in a memoir by Prouhet [17] in 1851, appears in various fields, such as combinatorics on words, symbolic dynamics, differential geometry, number theory, and mathematical physics, as surveyed by Allouche and Shallit [1]. That survey also mentions some generalizations of the Thue-Morse word, and recently other ones were considered in [2,8]. In particular, the following result was established in [2]. It was also shown in [2] that the word t b,m contains arbitrarily long squares, which extends a result previously established by Brlek [6] for m. Moreover, it was mentioned ( [2], p. 8) that, It would be interesting to determine the largest (fractional) power that occurs in the sequence t b,m . For b ≤ m, we already know that 2 is sharp.
We solve this problem here. Specifically, we study the family T of generalized Thue-Morse words consisting of the words t b,m as well as letter-renamings of them (see Section 2). For any t ∈ T , we compute the critical exponent (i.e., the supremum of the set of rational numbers that are exponents of powers in t) and determine exactly the occurrences of powers realizing it, both in terms of b and m. A noteworthy fact is that the critical exponents of generalized Thue-Morse words are always realized, which is not necessarily true in general (see Remarks 2.1 and 2.4).
The next section contains all of the basic terminology on words, borrowed mainly from Lothaire [14], along with the generalized Thue-Morse words. Section 3 contains the technical lemmas, establishing combinatorial properties used for proving our main results. The critical exponent (Theorem 4.4) is then computed in Section 4, and Section 5 deals with the occurrences of factors realizing it (Theorem 5.5).

Definitions and notation
Let Σ denote a finite alphabet, i.e., a finite set of symbols called letters. A finite word over Σ is a finite sequence w = w 0 w 1 · · · w ℓ−1 , where each w i ∈ Σ. We often write w[i] for w i , when i is a complicated formula. The length of w is |w| = ℓ, and the empty word of length 0 is denoted by ε. The set of all finite words over Σ is denoted by Σ * .
A word u is a factor of w if w = pus for some words p, s. Moreover, such a word u is said to be a proper factor of w if u = w. We also say that u is a prefix (resp. suffix) of w if p = ε (resp. s = ε). The number |p| is called an occurrence of u in w, i.e., |p| is the beginning position of an appearance of u in w. The set of all factors of a word w is denoted by F (w). A word v is a conjugate of w if there exists a word u such that uv = wu.
An overlap w is a word of the form w = auaua where a is a letter and u is a (possibly empty) word. For example, the English word alfalfa is an overlap and banana has the overlap anana as a suffix. A finite word is said to be overlap-free if it does not contain an overlap as a factor. Factors of the form uu are called squares.
The rational power of a word w is defined by w r = w ⌊r⌋ p where r is a rational such that r|w| ∈ N and p is the prefix of w of length (r − ⌊r⌋)|w|. For example, the word cab abbacc abbacc abb bca contains a factor which is a 5 2 -power. A (right) infinite word (or simply a sequence) x over Σ is a sequence indexed by N with values in Σ, i.e., x = x 0 x 1 x 2 · · · where each x i ∈ Σ. All of the terminology above naturally extends to infinite words.
An ultimately periodic infinite word can be written as uv ω = uvvv · · · , for some u, v ∈ Σ * , v = ε. If u = ε, then such a word is periodic. An infinite word that is not ultimately periodic is said to be aperiodic.
For any factor w of an infinite word x, the index of w in x is given by the number if such a number exists; otherwise, w is said to have infinite index in x. The critical exponent E(x) of an infinite word is given by It may be finite or infinite. A factor w of x is said to be a critical factor if its index realizes the critical exponent of x, that is, when Index(w) = E(x).
Remark 2.1. For any factor w of an infinite word x, Index(w) is always realized when finite, whereas the critical exponent E(x), even when finite, is not always realized; in particular, an infinite word may contain no critical factors. For example, the critical exponent of the well-known Fibonacci word f is is the golden ratio (see [15]), but none of the factors of f realize E(f ).

Generalized Thue-Morse words
There exist many generalizations of the Thue-Morse word. Here, we introduce a morphism based formulation which is more convenient for our purposes.
Hereafter, we study the family T of generalized Thue-Morse words t, for all b ≥ 2 , m ≥ 1 and α ∈ Σ.
This gives the following morphism : which is the case proposed by Allouche and Shallit [2].
where i ≡ i mod m. This is a subclass of the family of symmetric morphisms, defined by A. Frid [8]. That paper also contains an extension of Proposition 1.1 to more general words.
It is important to note that words in T , up to letter renaming, are exactly those given in Equation (1). Our definition avoids modular arithmetic on integers, and simplifies proofs by using the combinatorial properties of T instead. For that purpose, we say that a word w As a consequence, blocks of t are σ-cyclic.
Remark 2.4. Since t ∈ T is a fixpoint of a uniform non-erasing morphism, it follows immediately from Krieger's results in [13] that the critical exponent of t is either infinite (if t is periodic) or rational. Moreover, since µ(α) and µ(β) neither begin nor end with a common word when α = β, it also follows from [13] that the critical exponent, when finite, is reached (i.e., t contains critical factors).

Preliminary results
In this section, t ∈ T denotes an infinite generalized Thue-Morse word, as given in Definition 2.2.
Lemma 3.2. Let w be a σ-cyclic factor of t of length ℓ. If there exists an occurrence of w overlapping three consecutive blocks, then t is periodic.
Proof. Suppose that there exists such an occurrence of w in t. Since w is σ-cyclic and since every block is σ-cyclic, those three consecutive blocks, say β 1 , β 2 and β 3 , satisfy the fact that β 1 β 2 β 3 is σ-cyclic.
Proof. Let p be any prefix of length ℓ ′ of w and assume v = wp occurs in t.
Since w e = www e−2 , w is σ-cyclic from i). Let x be the conjugate of w such that w 0 x = ww 0 . Then, w e = w 0 xxx e−2−1/ℓ and from i) x is σ-cyclic and so too is its factor w ℓ−1 w 0 . Hence, w e is σ-cyclic, which ends the second part.
Proof. By contradiction, assume v = www 0 occurs in t, where w 0 is the first letter of w. Lemma 3.3 implies that v is σ-cyclic. Moreover, |v| = 2ℓ + 1 > 2b, which means that v overlaps at least three consecutive blocks of t. Thus, by Lemma 3.2, t is periodic which contradicts our assumption.

Critical exponent
In this section, we use exactly the same notation as previously. Before proving Theorem 4.4, we need a few additional facts.
The next three lemmas allow us to consider particular occurrences of factors of t. We say that an occurrence i of w in t is synchronized if b | i and b | (i+|w|).
Lemma 4.1. Suppose w = w 0 w 1 · · · w ℓ−1 is a factor of t such that b | ℓ. Let p be a non-empty prefix of w such that bq + r is an occurrence of w n p in t, where q, r ∈ N, 0 ≤ r < b and n ≥ 1. Let s be the possibly empty suffix of w such that w = ps. Moreover, let bq ′ + r ′ = bq + r + |w n p|, where q ′ , r ′ ∈ N, 0 ≤ r ′ < b. Then, iii) bq is a synchronized occurrence of vw n pu in t.
and let s = s 0 s 1 · · · s r−1 be the suffix of w of length r. We show that v = s. First note that vw 0 is contained in the block starting at position bq. Also, since b | ℓ, sw 0 is contained in one block. Both remarks imply that vw 0 and sw 0 are σ-cyclic. Hence, which gives the result. The proof of ii) is symmetric to the proof of i), and iii) follows from i) and ii). Proof. Let p be the non-empty prefix of w such that w e = w n p where n ∈ N + and let s be the suffix of w such that w = ps. Also, let bq + r be an occurrence of w e in t, where q, r ∈ N and 0 ≤ r < b. By Lemma 4.1, there exist a suffix v of w and a prefix u of sp such that bq is a synchronized occurrence of vw e u in t.
Now let y be the prefix of w such that w = yv, and define x = vy. We have vw e u = v(yv) n pu = (vy) n vpu = x n vpu. Moreover, vpu is a prefix of vpsps = vw 2 = v(yv) 2 = x 2 v so that vw e u = x n vpu = x f for some rational f . Finally, which ends the proof, since x and x f both satisfy the required conditions.
The next lemma deals with factors w of t of length ℓ not divisible by b.

Lemma 4.3.
Suppose t is aperiodic and let w be a factor of t of length ℓ such that b ∤ ℓ. Then Proof. We distinguish three cases according to ℓ, b and m.
Case 2 : ℓ < b and b > m. Let w 0 be the first letter of w and e = Index(w). If e > 2, then by Lemma 3.3 w e is σ-cyclic. In particular, w 0 = σ ℓ (w 0 ), i.e., m | ℓ and ℓ ≥ m. However, by Lemma 3.2, we know that |w e | ≤ 2b. Therefore, which ends this part.
Case 3 : ℓ < b and b ≤ m. As above, let e = Index(w) and suppose e > 2, which means w e is σ-cyclic and m | ℓ. The latter statement is not possible since 1 ≤ ℓ < b ≤ m. We conclude in this case that Index(w) ≤ 2.
We are now ready to prove the main theorem of this section, which gives the critical exponent of t.
Theorem 4.4. The critical exponent of t is given by Proof. If m | (b − 1), then E(t) = ∞ since t is periodic by Lemma 3.1. Now suppose m ∤ (b − 1) and let First we show that E(t) ≥ E b,m . If b ≤ m, it is easy to see that there is a square in the first two blocks of t, as noticed in [2]. On the other hand, if b > m, there exists a 2b/m-power in t. Indeed, let β 1 and β 2 be the blocks starting at positions b m − b and b m respectively. From Remark 2.3, the last letter of . Therefore, the whole factor β 1 β 2 of length 2b is σ-cyclic, and hence β 1 We now prove that E(t) = E b,m by showing that Index(w) ≤ E b,m for any factor w of t. Suppose w is a factor of t of length ℓ = b i N , where b ∤ N for some i, N ∈ N. The proof proceeds by induction on i.
Basis. If i = 0 then b ∤ ℓ. This case is proved in Lemma 4.3.
Hypothesis. We assume Index(w) ≤ E b,m for all factors w of t of length ℓ = b i N .
Induction. Let w be a factor of t of length ℓ = b i+1 N . Assume e = Index(w) > 1. Since b | ℓ, from Lemma 4.2 we know that there exists a factor x of length ℓ and a rational f ≥ e such that both x and x f are synchronized with the blocks. Let p be the proper prefix of x such that x f = x n p where n ∈ N + . Then, By the induction hypothesis, we have

Occurrences of critical factors
In this section, we assume that t is aperiodic, i.e., m ∤ (b − 1). We say that b > m is the overlap case and that b ≤ m is the square case and denote by e the critical exponent of t.
Here, we describe the occurrences and the lengths of the critical factors of t.
Lemma 5.1. Let w be a critical factor of t of length ℓ. Then the following properties hold.
i) µ(w) is a critical factor of t.
ii) If b | ℓ, then µ −1 (w) is a critical factor of t.
Proof. Property i) is trivial. For ii), suppose b | ℓ. If w e is not synchronized with the blocks, then Lemma 4.1 contradicts the maximality of e. Therefore, both w and w e are synchronized and their preimages under µ are well-defined. Finally, from inequality (3), we obtain the index of the preimage.
In view of Lemma 5.1, it is enough to consider only the case b ∤ ℓ when describing the occurences of critical factors of length ℓ in t.
Lemma 5.2. Let w be a critical factor of t of length ℓ such that b ∤ ℓ.
ii) In the square case, write w 2 = w (1) w (2) . Then w (2) occurs at the beginning of a block.
Proof. Overlap case (b > m). We consider separately the cases ℓ < b and ℓ > b.
If ℓ < b, then we have ℓ = m and |w e | = 2b by inequality (2). It follows that w e is composed of two consecutive blocks. Otherwise, if w e overlaps three consecutive blocks, then from Lemma 3.2 t is periodic, a contradiction.
On the other hand, if ℓ > b, then from the last section we know that Index(w) = 2b/m > 2. This contradicts Lemma 3.4; hence there is no such critical factor.
Square case (b ≤ m). Again, we distinguish two cases. First assume ℓ < b. Note that each block of t contains distinct letters. Also, w 2 is a factor of t and, from Lemma 3.3, w is σ-cyclic. If α i denotes the i-th letter of w 2 with 0 ≤ i < 2ℓ, then α i = α i+ℓ for all 0 ≤ i < ℓ. This implies that α 0 and α ℓ are not in the same block so that there exist two consecutive letters α i−1 and α i , where 1 ≤ i ≤ ℓ, which are not contained in the same block. Suppose 1 ≤ i < ℓ. Then, α i and α i+ℓ belong to the same block since ℓ < b. This is a contradiction because α i = α i+ℓ . We conclude that the pair α ℓ−1 and α ℓ is not contained in the same block. In other words, w (2) occurs at the beginning of a block.
Lastly, consider the case ℓ > b. From Lemma 3.3, w is σ-cyclic. Suppose w (2) does not occur at the beginning of a block. Then the pair w ℓ−1 w 0 formed by the last and the first letter of w is in the same block and is σ-cyclic. Hence, the whole factor w 2 of length 2ℓ > 2b is σ-cyclic. This factor overlaps three consecutive blocks so t is periodic from Lemma 3.2, a contradiction.
It has already been noticed that squares of certain factors of length 3 appear in the Thue-Morse word (see [6] for example). The following lemma proves the uniqueness of this fact. Proof. Suppose that w is critical factor of t of length ℓ > b such that b ∤ ℓ. In the proof of Lemma 5.2, we saw that in the overlap case there is no such critical factor. In the square case, we know also from Lemma 5.2 that if w 2 = w (1) w (2) then w (2) occurs at the beginning of a block. Moreover, Lemma 3.3 implies that w is σ-cyclic. Since ℓ > 2b implies that w overlaps three consecutive blocks, it follows from Lemma 3.2 that b < ℓ < 2b.
We now prove the following lemma which is a more general result than Lemma 5 in [2].
Lemma 5.4. Let k, N ∈ N + be such that b ∤ k and 1 ≤ N < b. Then, Proof. By direct computation, From Bezout's Identity, we know that for any x, y ∈ Z there exist s, t ∈ Z such that gcd(x, y) = sx + ty where s can be chosen positive. Let g ∈ Z and S g x,y = {s ∈ N + | g = sx + ty, t ∈ Z}.
The set S g x,y is non-empty when gcd(x, y) | g. Moreover, let us define the following three sets: We are now ready to state the main theorem of this section, which gives the set of occurrences where critical factors in an aperiodic generalized Thue-Morse word t realize the critical exponent e = E(t).
Theorem 5.5. If w is a critical factor of t of length ℓ = N b i such that b ∤ N , then the set of occurrences of w e in t is The proof of Theorem 5.5 follows easily from Lemma 5.1 and the next lemma. Lemma 5.6. Let w be a critical factor of t of length ℓ such that b ∤ ℓ. Then, the set of occurrences of w e in t is Proof. Overlap case (b > m). From Lemma 5.2, we know that w e = β 1 β 2 , where β 1 and β 2 are blocks. Suppose p is an occurence of w e and let kb q be the starting position of β 2 where b ∤ k and k, q ∈ N + . Then p = kb q − b. Now, let γ 1 = t[kb q − 1] be the last letter of β 1 and α 2 = t[kb q ] be the first letter of β 2 .
Since w e = β 1 β 2 is σ-cyclic from Lemma 3.3, we have That is, and hence m | q(b − 1). Therefore, there exists t ∈ Z such that mt = q(b − 1). In particular, we have m = q(b − 1) − (t − 1)m where q ∈ S m b−1,m which ends the first part.
Square case (b ≤ m). If we write w 2 = w (1) w (2) , then from Lemma 5.2 we know that w (2) occurs at the beginning of a block. We distinguish two cases.