The Critical Exponent is Computable for Automatic Sequences

Details The Critical Exponent is Computable for Automatic Sequences The Critical Exponent is Computable for Automatic Sequences

2011-04-12

arxiv.org/abs/1104.2303v3

EPTCS 63, 2011, pp. 231-239

Computer Science

Formal Languages and Automata Theory

In Proceedings WORDS 2011, arXiv:1108.3412

Scientific paper

10.4204/EPTCS.63.29

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.

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