On Pansiot Words Avoiding 3-Repetitions

TL;DR

This paper investigates the structure and growth of Pansiot words avoiding 3-repetitions over large alphabets, revealing their complexity growth approaches that of a ternary 2D language with an estimated rate of 1.2421.

Contribution

It proves that the complexity growth rates of these regular languages tend to a specific value as alphabet size increases, linking one-dimensional and two-dimensional language complexities.

Findings

Growth rates approach 1.2421 as alphabet size increases

Pansiot words avoiding 3-repetitions form a regular language for k >= 5

Complexity of these languages relates to ternary 2D language complexity

Abstract

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 about 1.2421.