The repetition threshold for binary rich words

TL;DR

This paper proves that the minimal repetition threshold for infinite binary rich words is exactly 2+√2/2, confirming a conjecture and resolving an open problem for binary alphabets.

Contribution

It provides a structure theorem for binary rich words avoiding high-exponent repetitions, establishing the exact repetition threshold for binary rich words.

Findings

Repetition threshold for binary rich words is 2+√2/2.

Binary rich words avoiding 14/5-power repetitions are characterized.

Confirms the conjecture by Baranwal and Shallit.

Abstract

A word of length $n$ is rich if it contains $n$ nonempty palindromic factors. An infinite word is rich if all of its finite factors are rich. Baranwal and Shallit produced an infinite binary rich word with critical exponent $2+\sqrt{2}/2$ ($\approx 2.707$) and conjectured that this was the least possible critical exponent for infinite binary rich words (i.e., that the repetition threshold for binary rich words is $2+\sqrt{2}/2$). In this article, we give a structure theorem for infinite binary rich words that avoid $14/5$-powers (i.e., repetitions with exponent at least 2.8). As a consequence, we deduce that the repetition threshold for binary rich words is $2+\sqrt{2}/2$, as conjectured by Baranwal and Shallit. This resolves an open problem of Vesti for the binary alphabet; the problem remains open for larger alphabets.