Circular repetition thresholds on some small alphabets: Last cases of Gorbunova's conjecture

TL;DR

This paper determines the exact values of the strong circular repetition threshold for 4 and 5-letter alphabets, confirming Gorbunova's conjecture, and establishes the intermediate and weak thresholds for 3-letter alphabets.

Contribution

It proves the last unknown values of the strong circular repetition threshold, confirming Gorbunova's conjecture, and equates the intermediate and weak thresholds with the classical repetition threshold for 3-letter alphabets.

Findings

CRT_S(4)=3/2

CRT_S(5)=4/3

CRT_I(3)=CRT_W(3)=7/4

Abstract

A word is called $\beta$-free if it has no factors of exponent greater than or equal to $\beta$. The repetition threshold $\mathrm{RT}(k)$ is the infimum of the set of all $\beta$ such that there are arbitrarily long $k$-ary $\beta$-free words (or equivalently, there are $k$-ary $\beta$-free words of every sufficiently large length, or even every length). These three equivalent definitions of the repetition threshold give rise to three natural definitions of a repetition threshold for circular words. The infimum of the set of all $\beta$ such that - there are arbitrarily long $k$-ary $\beta$-free circular words is called the weak circular repetition threshold, denoted $\mathrm{CRT}_{\mathrm{W}}(k)$; - there are $k$-ary $\beta$-free circular words of every sufficiently large length is called the intermediate circular repetition threshold, denoted $\mathrm{CRT}_{\mathrm{I}}(k)$; - there are $k$-ary $\beta$-free circular words of every length is called the strong circular repetition threshold, denoted $\mathrm{CRT}_{\mathrm{S}}(k)$. We prove that $\mathrm{CRT}_{\mathrm{S}}(4)=\tfrac{3}{2}$ and $\mathrm{CRT}_{\mathrm{S}}(5)=\tfrac{4}{3}$, confirming a conjecture of Gorbunova and providing the last unknown values of the strong circular repetition threshold. We also prove that $\mathrm{CRT}_{\mathrm{I}}(3)=\mathrm{CRT}_{\mathrm{W}}(3)=\mathrm{RT}(3)=\tfrac{7}{4}$.