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Infinite words containing squares at every position

James CurrieNarad Rampersad — 2010

RAIRO - Theoretical Informatics and Applications

Richomme asked the following question: what is the infimum of the real numbers > 2 such that there exists an infinite word that avoids -powers but contains arbitrarily large squares beginning at every position? We resolve this question in the case of a binary alphabet by showing that the answer is = 7/3.

Squares and overlaps in the Thue-Morse sequence and some variants

Shandy BrownNarad RampersadJeffrey ShallitTroy Vasiga — 2006

RAIRO - Theoretical Informatics and Applications

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.

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