On Pansiot Words Avoiding 3-Repetitions

A (1-or 2-dimensional) language is factorial, if it is closed under taking factors of its words. A word w avoids a word u if u is not a factor of w. The set of all minimal (with respect to the factor order) words avoided by all elements of a factorial language L is called the antidictionary of L. All 1-dimensional languages with finite antidictionaries are regular.

We denote the antidictionary of the threshold language T k by A k . A word u ∈ A k can be factorized as u = yzy, where |yz| = per(u), |u|/|yz| > k/(k-1), and all proper factors of u have the exponent at most k/(k-1). If |y| = m, we call u an m-repetition.

The finite set A k . Since an infinite regular language contains arbitrary powers of some word, one has

k . The combinatorial complexity of a language L is a function C L (n) which returns the number of words in L of length n. This function serves as a natural quantitative measure of L. “Big” [“small”] languages have exponential [resp., subexponential] complexity. Exponential complexity can be described by means of the growth rate α(L) = lim sup n→∞ (C L (n)) 1/n (subexponential complexity is indicated by α(L) = 1). For factorial languages, classical Fekete’s lemma implies

approximates the growth rate of T k from above. It is easy to prove that lim m→∞ α(T

For regular languages, the growth rate equals the index (spectral radius of the adjacency matrix) of recognizing automaton, providing that this automaton is consistent (each vertex belongs to some accepting walk), and either deterministic, or non-deterministic but unambiguous (there is at most one walk with the given label between two given vertices); see [13].

In [10], Pansiot showed how to encode all words from the language T (2) k with “characteristic” words over the alphabet {0, 1}. This encoding played a big role in the proof of Dejean’s conjecture; so, we refer to the elements of T

(2) k as to Pansiot words. These words can be equivalently defined by the following pair of conditions: (P1) two closest occurrences of a letter are on the distance k-1, k, or k+1;

(P2) two closest occurrences of a letter are followed by different letters.

We also consider Pansiot Z-words, which are given by (P1), (P2) as well. Finite factors of Pansiot Z-words are exactly Pansiot words. Now we introduce cylindric representation of Pansiot words. Imagine such a word (finite or infinite) as a rope with knots, which are representing letters. This rope is wound around a cylinder such that the knots at distance k are placed one under another (Fig. 1,a). By (P1), the knots labeled by two closest occurrences of the same letter appear on two consecutive winds of the rope one under another or shifted by one knot (Fig. 1,b). If we connect these closest occurrences by “sticks”, we get three types of such sticks: vertical, left-slanted, and right-slanted (Fig. 1,b). We associate each letter in a Pansiot word with a stick going up from the corresponding knot, getting an encoding of this word by a cylindric word over the ternary alphabet ∆ = { , , }. to the permutation of the alphabet. Note that cylindric words avoid squares of letters in view of (P2). Hence, cylindric Z-words are just infinite sequences of blocks and . The feature of cylindric words is that they have an additional 2-dimensional structure, allowing one to capture structural properties of Pansiot words through 2-dimensional factors of cylindric words. We say that a Z-word W is compatible to a language L if all factors of W belong to L. Theorem 1 ( [12]). For any integer m ≥ 3, there exists a set S m of 2-dimensional words of size

This theorem states that cylindric words that encode the words from T (m) k are defined by 2dimensional avoidance properties. For example, cylindric words of the Pansiot words avoiding 3repetitions are defined by the avoidance of the structures and . Indeed, any of these structures implies the existence of three successive letters (say, a, b, and c) in the encoded Pansiot word such that two occurrences of the factor abc appear one under another at the distance 2k; since (2k+3)/2k > k/(k-1), the encoded word contains a 3-repetition.

For a language L, let L be its subset consisting of all factors of Z-words compatible to L. By [14, Theorem 3.1], α( L) = α(L). Let Cyl

Thus, the growth rates of threshold languages can be estimated through the study of cylindric words with simple avoidance properties that are independent of the size of the alphabet. In what follows, we refer to the elements of Cyl k )} ∞ 5 has a limit as k approaches infinity, and this limit is the “growth rate” of the 2-dimensional language defined by the same avoidance properties as Cyl (m) k . Through the computations of growth rates for the alphabets with 5, 6, . . . , 60 letters we observed in [12] that the sequence {α(T (3) k )} demonstrates fast convergence

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