Squares and overlaps in the Thue-Morse sequence and some variants
We consider the position and number of occurrences of squares in the Thue-Morse sequence, and show that the corresponding sequences are -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 ) contains an overlap.