Gaps in the Thue--Morse word

TL;DR

This paper investigates the gap sequence between occurrences of factors in the Thue--Morse sequence, proving it is non-automatic for factors of length at least two, and provides methods to compute discrepancy sequences.

Contribution

It proves the non-automaticity of the gap sequence for factors of length ≥2 and introduces a method to compute discrepancy sequences using transducers.

Findings

The gap sequence is morphic but not automatic for factors of length ≥2.

An explicit method to compute discrepancy of '01' occurrences is provided.

Discrepancy sequences correspond to output sums of a specific base-2 transducer.

Abstract

The Thue--Morse sequence is a prototypical automatic sequence found in diverse areas of mathematics, and in computer science. We study occurrences of factors $w$ within this sequence, more precisely, the sequence of gaps between consecutive occurrences. This gap sequence is morphic; we prove that it is not automatic as soon as the length of $w$ is at least two, thereby answering a question by J.~Shallit in the affirmative. We give an explicit method to compute the \emph{discrepancy} of the number of occurrences of the block $\mathtt{01}$ in the Thue--Morse sequence. We prove that the sequence of discrepancies is the sequence of output sums of a certain base-$2$ transducer.