The Critical Exponent is Computable for Automatic Sequences

TL;DR

This paper proves that the critical exponent of k-automatic sequences is always rational or infinite and can be computed, extending previous results and applying to related exponents and recurrence constants.

Contribution

It establishes the computability and rationality of the critical exponent for k-automatic sequences, generalizing prior work and covering variants like the Diophantine and initial critical exponents.

Findings

Critical exponent is always rational or infinite for k-automatic sequences.

The critical exponent can be effectively computed for these sequences.

Results extend to variants like the Diophantine and initial critical exponents.

Abstract

The critical exponent of an infinite word is defined to be the supremum of the exponent of each of its factors. For k-automatic sequences, we show that this critical exponent is always either a rational number or infinite, and its value is computable. Our results also apply to variants of the critical exponent, such as the initial critical exponent of Berthe, Holton, and Zamboni and the Diophantine exponent of Adamczewski and Bugeaud. Our work generalizes or recovers previous results of Krieger and others, and is applicable to other situations; e.g., the computation of the optimal recurrence constant for a linearly recurrent k-automatic sequence.