
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 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 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 in the Thue--Morse sequence. We prove that the sequence of discrepancies is the sequence of output sums of a certain base- transducer.
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