Gaps in the Thue--Morse word
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.
Cite
@article{arxiv.2102.01018,
title = {Gaps in the Thue--Morse word},
author = {Lukas Spiegelhofer},
journal= {arXiv preprint arXiv:2102.01018},
year = {2021}
}
Comments
36 pages. Accepted for publication in J. Aust. Math. Soc. This version is a significant extension of v1