Transition Property for Cube-Free Words

Abstract

We study cube-free words over arbitrary non-unary finite alphabets and prove the following structural property: for every pair (u, v) of d-ary cube-free words, if u can be infinitely extended to the right and v can be infinitely extended to the left respecting the cube-freeness property, then there exists a "transition" word w over the same alphabet such that uwv is cube free. The crucial case is the case of the binary alphabet, analyzed in the central part of the paper.The obtained "transition property", together with the developed technique, allowed us to solve cube-free versions of three old open problems by Restivo and Salemi. Besides, it has some further implications for combinatorics on words; e.g., it implies the existence of infinite cube-free words of very big subword (factor) complexity.Problem 2. Given an α-power-free word u, construct explicitly an α-power-free infinite word having u as prefix, provided that u is right extendable. Problem 3. Given an integer k ≥ 0, does there exist an α-power-free word u with the properties (i) there exists a word v of length k such that uv is α-power free and (ii) for every word v of bigger length, uv is not α-power free. Problem 4. Given two α-power-free words u and v, decide whether there is a "transition" from u to v (i.e., does there exist a word w such that uwv is α-power free). Problem 5. Given two α-power-free words u and v, find explicitly a transition word w, if it exists.These natural problems appear to be rather hard. Only for Problem 1a,b there is a sort of a general solution: a backtracking decision procedure exists for all k-power-free languages, where k ≥ 2 is an integer [3,4]. In a number of cases, the parameters of backtracking were found by computer search, so it is not clear whether this technique can be extended for α-power-free words with rational α. The decision procedure also gives no clue to Problem 2.There is a particular case of binary overlap-free words, for which all problems are solved in [1,15] (more efficient solutions were given in [2]). These words have a regular structure deeply related to the famous Thue-Morse word, and it seems that all natural algorithmic problems for them are solved. For example, the asymptotic order of growth for the binary overlap-free language is computed exactly [6,7], and even the word problem in the corresponding syntactic monoid has a linear-time solution [18]. Most of the results can be extended, with additional technicalities, to binary α-power-free words for any α ≤ 7/3, because the structure of these words is essentially the same as of overlap-free words (see, e.g., [8]). However, the situation changes completely if we go beyond the polynomial-size language of binary (7/3)-power-free words. In the exponential-size α-power-free languages 3 the diversity of words is much bigger, so it becomes harder to find a universal decision procedure. The only results on Problems 1-5 apart from those mentioned above are the positive answers to Problem 3 (including its two-sided analog) for the two classical test cas...

Branching Densities of Cube-Free and Square-Free Words

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Transition Property for $α$-Power Free Languages with $α\geq 2$ and $k\geq 3$ Letters

“…We define the repetition threshold RT(k) to be the infimum of all rational numbers α such that there exists an infinite α-power-free word over an alphabet with k letters. Dejean's conjecture states that RT(2) = 2, RT(3) = 7 4 , RT(4) = 7 5 , and RT(k) = k k−1 for each k > 4 [3]. Dejean's conjecture has been proved by the work of several authors [1,2,3,5,6,9].…”

Construction of a bi-infinite power free word with a given factor and a non-recurrent letter

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“…In particular, the problems 4 and 5 are solved for some overlap free (2 + -power free) binary words. In addition, in [1] the authors prove that: For every pair (u, v) of cube free words (3-power free) over an alphabet with k letters, if u can be infinitely extended to the right and v can be infinitely extended to the left respecting the cube-freeness property, then there exists a "transition" word w over the same alphabet such that uwv is cube free. Let N denote the set of positive integers and let Q denote the set of rational numbers.…”