Squares and overlaps in the Thue-Morse sequence and some variants
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Shandy Brown, Narad Rampersad, Jeffrey Shallit, and Troy Vasiga, "Squares and overlaps in the Thue-Morse sequence and some variants." RAIRO - Theoretical Informatics and Applications / Informatique Théorique et Applications 40 (2006): 473-484. DOI: 10.1051/ita:2006030.
We consider the position and number of occurrences of squares in the Thue-Morse sequence, and show that the corresponding sequences are 2-regular. We also prove that changing any finite but nonzero number of bits in the Thue-Morse sequence creates an overlap, and any linear subsequence of the Thue-Morse sequence (except those corresponding to decimation by a power of 2) contains an overlap.