Papers › On monochromatic arithmetic progressions in binary words associated with pattern sequences
On monochromatic arithmetic progressions in binary words associated with pattern sequences
Bartosz Sobolewski
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Let eᵥ(n) denote the number of occurrences of a fixed pattern v in the binary expansion of n ∈ℕ. In this paper we study monochromatic arithmetic progressions in the class of binary words (eᵥ(n) 2)_(n ≥0), which includes the famous Thue--Morse word 𝐭 and Rudin--Shapiro word 𝐫. We prove that the length of a monochromatic arithmetic progression of difference d ≥3 starting at $0$ in 𝐫 is at most (d+3)/2, with equality for infinitely many d. Moreover, we compute the maximal length of a monochromatic arithmetic progression in 𝐫 of difference 2ᵏ-1 and 2ᵏ+1. For a general pattern v we provide an upper bound on the length of a monochromatic arithmetic progression of any difference d. We also prove other miscellaneous results and offer a number of related problems and conjectures.
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