Gaps in the Thue-Morse word

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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, or 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 2 , thereby answering a question by J. Shallit in the affirmative. We give an explicit method to compute the discrepancy of the number of occurrences of the block 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.
Original languageEnglish
Pages (from-to)110-144
Number of pages35
JournalJournal of the Australian Mathematical Society
Issue number1
Publication statusPublished - 25 Feb 2023

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Publisher Copyright: © The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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