On balanced sequences and their critical exponent
Abstract
We study aperiodic balanced sequences over finite alphabets. A sequence vv of this type is fully characterised by a Sturmian sequence u and two constant gap sequences y and y'. We show that the language of v is eventually dendric and we focus on return words to its factors. We develop a method for computing the critical exponent and asymptotic critical exponent of balanced sequences, provided the associated Sturmian sequence u has a quadratic slope. The method is based on looking for the shortest return words to bispecial factors in v. We illustrate our method on several examples; in particular we confirm a conjecture of Rampersad, Shallit and Vandomme that two specific sequences have the least critical exponent among all balanced sequences over 9-letter (resp., $0-letter) alphabets.
Cite
@article{arxiv.2108.07503,
title = {On balanced sequences and their critical exponent},
author = {Francesco Dolce and Lubomira Dvorakova and Edita Pelantova},
journal= {arXiv preprint arXiv:2108.07503},
year = {2022}
}