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.

Quadratic approximation to automatic continued fractions

“…Likewise, we may suspect that the set of values taken by w 2 , w * 2 and w 2 − w * 2 at automatic numbers does not include irrational numbers. All this would be supported by a result of Schaeffer and Shallit [28] on the Diophantine exponent of automatic sequences, recalled at the end of Section 2.…”

“…We conclude this section with an important result of Schaeffer and Shallit [28], already alluded to in Section 1.…”

Undecidability and Finite Automata

“…Recently the second author and co-authors defined a notation of k-automaticity for sets of non-negative rational numbers [12,10], in analogy with the more wellknown concept for sets of non-negative integers [3]. For an integer k ≥ 2 define Σ k = {0, 1, .…”

Subword Complexity and k-Synchronization

“…We already showed how to construct an automaton accepting {(n, ρ x (n)) k : n ≥ 1}. Now we just use the results from [17,16]. Notice that the lim sup corresponds to what is called the largest "special point" in [16].…”