The Critical Exponent is Computable for Automatic Sequences

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 Berthé, 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.

Enumeration and Decidable Properties of Automatic Sequences

“…Finally, in a recent paper [34], the third author shows that additional properties of automatic sequences are deducible by expanding on the techniques in this paper. For example, the critical exponent is computable.…”

The Critical Exponent Is Computable for Automatic Sequences

“…In a previous version of this paper that was presented at WORDS 2011, the second author stated that if L is regular, then lim sup x∈L quo k (x) is either rational or infinite, and is computable [21]. Unfortunately the envisioned proof of this more general result contained a subtle flaw that we have not been able to repair.…”

“…A preliminary version of this paper was presented at the WORDS 2011 conference in Prague, Czech Republic [21].…”

Primitive Words and Lyndon Words in Automatic and Linearly Recurrent Sequences

“…For more details, see [5] or [3]. In several previous papers [1,4,17,19,11], we have developed a technique to show that many properties of automatic sequences are decidable. The fundamental tool is the following: Theorem 1.…”