Upper bound for the generalized repetition threshold

TL;DR

This paper establishes bounds on the generalized repetition threshold for strings over an alphabet, showing it is tightly constrained between two functions of alphabet size and minimal factor length.

Contribution

It provides new upper and lower bounds for the fractional power threshold in strings, extending previous definitions and understanding of repetition limits.

Findings

Proves that R(a,l) is at least 1 + 1/(la).

Shows that R(a,l) is at most 1 + c/(la) for some constant c.

Establishes that the generalized repetition threshold is tightly bounded by these functions.

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 $x^{r'}$ where $x$ has length at least $l$ and $r'\ge r$. We prove that $1+\frac{1}{la}\le R(a,l)\le 1+\frac{c}{la}$ for some constant $c$.