Dejean's conjecture holds for N ≥ 27

Currie, James D.; Rampersad, Narad

doi:10.1051/ita/2009017

Cited by 22 publications

(41 citation statements)

References 13 publications

(32 reference statements)

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“…Thue, Dejean and Pansiot, respectively [19,6,14], established the values RT (2), RT (3), RT (4). Moulin Ollagnier [13] verified Dejean's conjecture for 5 ≤ n ≤ 11, and Mohammad-Noori and Currie [12] proved the conjecture for 12 ≤ n ≤ 14.…”

Section: Introductionmentioning

confidence: 97%

A proof of Dejean’s conjecture

Currie

Rampersad

2011

Math. Comp.

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Section: Introductionmentioning

confidence: 97%

A proof of Dejean’s conjecture

Currie

Rampersad

2011

Math. Comp.

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“…. , 60 letters we observed in [12] that the sequence {α(T (3) k )} demonstrates fast convergence to the limit ≈1.242096777.…”

Section: Theorem 1 ( [12]) For Any Integer M ≥ 3 There Exists a Setmentioning

confidence: 74%

“…Deterministic cover automaton is a special case of unambiguous nondeterministic automaton; so, the index of R equals α(Cyl (3) k ). Now consider the kth power R k of R. Note that in most cases the correctness of transition from some vertex u of R k to some other vertex v can be checked using only k last symbols of u.…”

Section: Automatamentioning

confidence: 99%

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On Pansiot Words Avoiding 3-Repetitions

Gorbunova

Shur

2012

Int. J. Found. Comput. Sci.

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The recently confirmed Dejean's conjecture about the threshold between avoidable and unavoidable powers of words gave rise to interesting and challenging problems on the structure and growth of threshold words. Over any finite alphabet with k ≥ 5 letters, Pansiot words avoiding 3-repetitions form a regular language, which is a rather small superset of the set of all threshold words. Using cylindric and 2-dimensional words, we prove that, as k approaches infinity, the growth rates of complexity for these regular languages tend to the growth rate of complexity of some ternary 2-dimensional language. The numerical estimate of this growth rate is ≈1.2421.Powers, integral and fractional, are the simplest and most natural repetitions in words. Any repetition over an arbitrary fixed alphabet is characterized by the set of all words over this alphabet, avoiding this repetition. The main question concerning such a set is whether it is finite or infinite. For fractional powers, this question is answered by Dejean's conjecture [5], which is now proved in all cases by the efforts of different authors, see [2][3][4][8][9][10][11].Recall that the exponent of a word w is the ratio between its length and its minimal period: exp(w) = |w|/ per(w). If exp(w) = β > 1, then w is a fractional power (β -power). It is convenient to treat the notion of β -power as follows: a word w is a β -power if exp(w) ≥ β while (|w|−1)/ per(w) < β , and a β + -power if exp(w) > β while (|w|−1)/ per(w) ≤ β . As usual, β + is treated as a "number", covering β in the usual ≤ order. A word is called β -free (where β can be a number with plus as well) if it contains no β -powers as factors. A β -power is k-avoidable if the number of k-ary β -free words is infinite. Dejean's conjecture states that a β -power is k-avoidable if and only if β ≥ (7/4) + and k = 3, β ≥ (7/5) + and k = 4, or β ≥ (k/(k−1)) + and k = 2, k ≥ 5.The (k/(k−1)) + -free languages over k-letter alphabets, where k ≥ 5, are called threshold languages; we denote them by T k . We study structure and growth of these languages, aiming at the asymptotic properties as the size of the alphabet increases. Any threshold language can be approximated from above by a series of regular languages consisting of words that locally satisfy the (k/(k−1)) + -freeness property. Namely, these words avoid all (k/(k−1)) + -powers w such that |w| − per(w) ≤ m, for some constant m. From our previous work [12], it is clear that the case m = 3 gives a lot of important structural information about the languages T k . Here we study this case in details, using cylindric representation that captures the properties common for considered words over all alphabets. PreliminariesWe study finite words and two-sided infinite words (Z-words) over finite k-letter alphabets Σ k and over some special ternary alphabet introduced below. We also consider 2-dimensional words, which are just finite rectangular arrays of alphabetic symbols. Unlike to some commonly used models of 2-dimensional

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“…Enumerating the repetitions in infinite words is a classic problem in combinatorics on words that has been studied in depth over the last 100 years (see for example, [10,3] and references therein).…”

Section: Introductionmentioning

confidence: 99%