Least periods of k-automatic sequences

TL;DR

This paper proves that the sequence of least periods of any k-automatic sequence is itself k-automatic, extending previous results and confirming them through implementation for several well-known sequences.

Contribution

It generalizes the concept of least periods to k-automatic sequences and demonstrates that their characteristic sequences are also k-automatic, providing an effective construction.

Findings

Confirmed that least periods sequence of Thue-Morse is k-automatic

Extended results to period-doubling, Rudin-Shapiro, and paperfolding sequences

Provided an implementation confirming theoretical results

Abstract

Currie and Saari initiated the study of least periods of infinite words, and they showed that every integer n >= 1 is a least period of the Thue-Morse sequence. We generalize this result to show that the characteristic sequence of least periods of a k-automatic sequence is (effectively) k-automatic. Through an implementation of our construction, we confirm the result of Currie and Saari, and we obtain similar results for the period-doubling sequence, the Rudin-Shapiro sequence, and the paperfolding sequence.