Repetition avoidance in products of factors

TL;DR

This paper investigates a variation of a classical combinatorics on words problem, determining the minimal repetition thresholds for products of factors in infinite words over a 3-letter alphabet, extending previous results.

Contribution

It provides exact formulas for the repetition threshold _i(3) for all integers i, generalizing earlier known cases and resolving open questions.

Findings

_i(3)= frac{3i}{2}+ frac{1}{4} for even i

_i(3)= frac{3i}{2}+rac{1}{6} for odd i  3

Extends known thresholds _i(2)=2i and _2(3)=13/4

Abstract

We consider a variation on a classical avoidance problem from combinatorics on words that has been introduced by Mousavi and Shallit at DLT 2013. Let $\texttt{pexp}_i(w)$ be the supremum of the exponent over the products of $i$ factors of the word $w$. The repetition threshold $\texttt{RT}_i(k)$ is then the infimum of $\texttt{pexp}_i(w)$ over all words $w\in\Sigma^\omega_k$. Mousavi and Shallit obtained that $\texttt{RT}_i(2)=2i$ and $\texttt{RT}_2(3)=\tfrac{13}4$. We show that $\texttt{RT}_i(3)=\tfrac{3i}2+\tfrac14$ if $i$ is even and $\texttt{RT}_i(3)=\tfrac{3i}2+\tfrac16$ if $i$ is odd and $i\ge3$.