Finite-Repetition threshold for infinite ternary words

TL;DR

This paper investigates the finite-repetition threshold for infinite words over ternary alphabets, establishing exact values for three-letter alphabets and conjecturing results for larger alphabets, advancing understanding of repetition constraints.

Contribution

It precisely determines the finite-repetition threshold for ternary alphabets and proposes conjectures for larger alphabets, refining the understanding of repetition limits in infinite words.

Findings

FRt(3)=7/4, equal to r(3), with minimal 2 factors

Achieved the bound with an infinite word containing only two 7/4-exponent factors

Conjecture that FRt(4)=7/5, remains open for larger alphabets

Abstract

The exponent of a word is the ratio of its length over its smallest period. The repetitive threshold r(a) of an a-letter alphabet is the smallest rational number for which there exists an infinite word whose finite factors have exponent at most r(a). This notion was introduced in 1972 by Dejean who gave the exact values of r(a) for every alphabet size a as it has been eventually proved in 2009. The finite-repetition threshold for an a-letter alphabet refines the above notion. It is the smallest rational number FRt(a) for which there exists an infinite word whose finite factors have exponent at most FRt(a) and that contains a finite number of factors with exponent r(a). It is known from Shallit (2008) that FRt(2)=7/3. With each finite-repetition threshold is associated the smallest number of r(a)-exponent factors that can be found in the corresponding infinite word. It has been proved by Badkobeh and Crochemore (2010) that this number is 12 for infinite binary words whose maximal exponent is 7/3. We show that FRt(3)=r(3)=7/4 and that the bound is achieved with an infinite word containing only two 7/4-exponent words, the smallest number. Based on deep experiments we conjecture that FRt(4)=r(4)=7/5. The question remains open for alphabets with more than four letters. Keywords: combinatorics on words, repetition, repeat, word powers, word exponent, repetition threshold, pattern avoidability, word morphisms.