On monochromatic arithmetic progressions in binary words associated with pattern sequences

Let $e_v\left(n\right)$ denote the number of occurrences of a fixed pattern v in the binary expansion of $n\in N$. In this paper we study monochromatic arithmetic progressions in the class of binary words ${\left(e_v\right(n\left)\mathrm{mod}2\right)}_{n\ge 0}$, 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 $d\ge 3$ starting at 0 in r is at most $(d+3)/2$, with equality for infinitely many d. We also compute the maximal length of a monochromatic arithmetic progression in r of difference $2^k-1$ and $2^k+1$. For a general pattern v we show that the maximal length of a monochromatic arithmetic progression of difference d is at most linear in d. Moreover, we prove that a linear lower bound holds for suitable subsequences ${\left(d_k\right)}_{k\ge 0}$ of differences. We also offer a number of related problems and conjectures.