In 2017, Vesti proposed the problem of determining the repetition threshold for infinite rich words, i.e., for infinite words in which all factors of length $n$ contain $n$ distinct nonempty palindromic factors. In 2020, Currie, Mol, and Rampersad proved a conjecture of Baranwal and Shallit that the repetition threshold for binary rich words is $2 + \sqrt{2}/2$. In this paper, we prove a structure theorem for $16/7$-power-free ternary rich words. Using the structure theorem, we deduce that the repetition threshold for ternary rich words is $1 + 1/(3 - \mu) \approx 2.25876324$, where $\mu$ is the unique real root of the polynomial $x^3 - 2x^2 - 1$.
@article{10_37236_13499,
author = {James Currie and Lucas Mol and Jarkko Peltom\"aki},
title = {The repetition threshold for ternary rich words},
journal = {The electronic journal of combinatorics},
year = {2025},
volume = {32},
number = {2},
doi = {10.37236/13499},
zbl = {8062189},
url = {http://geodesic.mathdoc.fr/articles/10.37236/13499/}
}
TY - JOUR
AU - James Currie
AU - Lucas Mol
AU - Jarkko Peltomäki
TI - The repetition threshold for ternary rich words
JO - The electronic journal of combinatorics
PY - 2025
VL - 32
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/13499/
DO - 10.37236/13499
ID - 10_37236_13499
ER -
%0 Journal Article
%A James Currie
%A Lucas Mol
%A Jarkko Peltomäki
%T The repetition threshold for ternary rich words
%J The electronic journal of combinatorics
%D 2025
%V 32
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/13499/
%R 10.37236/13499
%F 10_37236_13499
James Currie; Lucas Mol; Jarkko Peltomäki. The repetition threshold for ternary rich words. The electronic journal of combinatorics, Tome 32 (2025) no. 2. doi: 10.37236/13499
