The limit of repetition thresholds of rich sequences

Abstract

The repetition threshold of a class of sequences is the smallest number $r$ such that a sequence from the class contains no repetition with exponent $>r$. We focus on the class ${\mathcal{C}}_d$ of $d$-ary sequences rich in palindromes. In 2020, Currie, Mol, and Rampersad determined the repetition threshold for ${\mathcal{C}}_2$. In 2024, Currie, Mol and Peltomäki found the repetition threshold for ${\mathcal{C}}_3$ and conjectured that the repetition threshold for ${\mathcal{C}}_d$ tends to 2 with $d$ growing to infinity. Here we verify their conjecture.