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.