The Weak Circular Repetition Threshold Over Large Alphabets

TL;DR

This paper proves that the weak circular repetition threshold equals the repetition threshold for all large alphabets (n ≥ 45), confirming a significant part of a conjecture relating different repetition thresholds.

Contribution

It establishes that the weak circular repetition threshold matches the repetition threshold for all sufficiently large alphabets, specifically for n ≥ 45, advancing understanding of repetition properties in words.

Findings

Proves CRT_W(n) = RT(n) for all n ≥ 45

Confirms a weak version of the conjecture for all but finitely many n

Provides insight into repetition thresholds in combinatorics on words

Abstract

The repetition threshold for words on $n$ letters, denoted $\mbox{RT}(n)$, is the infimum of the set of all $r$ such that there are arbitrarily long $r$-free words over $n$ letters. A repetition threshold for circular words on $n$ letters can be defined in three natural ways, which gives rise to the weak, intermediate, and strong circular repetition thresholds for $n$ letters, denoted $\mbox{CRT}_{\mbox{W}}(n)$, $\mbox{CRT}_{\mbox{I}}(n)$, and $\mbox{CRT}_{\mbox{S}}(n)$, respectively. Currie and the present authors conjectured that $\mbox{CRT}_{\mbox{I}}(n)=\mbox{CRT}_{\mbox{W}}(n)=\mbox{RT}(n)$ for all $n\geq 4$. We prove that $\mbox{CRT}_{\mbox{W}}(n)=\mbox{RT}(n)$ for all $n\geq 45$, which confirms a weak version of this conjecture for all but finitely many values of $n$.