## Currie, James D. and Mol, Lucas and Rampersad, Narad - The repetition threshold for binary rich words

dmtcs:5791 - Discrete Mathematics & Theoretical Computer Science, February 24, 2020, vol. 22 no. 1 - https://doi.org/10.23638/DMTCS-22-1-6
The repetition threshold for binary rich words

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.

Volume: vol. 22 no. 1
Section: Analysis of Algorithms
Published on: February 24, 2020
Submitted on: September 27, 2019
Keywords: Mathematics - Combinatorics,Computer Science - Formal Languages and Automata Theory,68R15