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Gaps in the Thue--Morse word

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arxiv 2102.01018 v2 pith:R624ZJ2N submitted 2021-02-01 math.CO math.NT

Lukas Spiegelhofer This is my paper · ORCID
classification math.COmath.NT
keywords sequenceoccurrencesthue--morseautomaticgapsproveaffirmativeanswering
verification ladder T0 review T1 audit T2 compute T3 formal
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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.

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