Upper bound for the generalized repetition threshold

Abstract

Let $A$ be an $a$-letter alphabet. We consider fractional powers of $A$-strings: if $x$ is a $n$-letter string, $x^r$ is a prefix of $xxxx...$ having length $nr$. Let $l$ be a positive integer. Ilie, Ochem and Shallit defined $R(a,l)$ as the infimum of reals $r>1$ such that there exist a sequence of $A$-letters without factors (substrings) that are fractional powers $x^{r^{\prime }}$ where $x$ has length at least $l$ and $r^{\prime }\ge r$. We prove that $1+\frac{1}{la}\le R(a,l)\le 1+\frac{c}{la}$ for some constant $c$.