On monochromatic arithmetic progressions in binary words associated with pattern sequences

TL;DR

This paper investigates monochromatic arithmetic progressions in binary words derived from pattern occurrence counts modulo 2, including famous sequences like Thue–Morse and Rudin–Shapiro, providing bounds and exact lengths for such progressions.

Contribution

It establishes bounds on the length of monochromatic arithmetic progressions in these binary sequences and computes exact maximal lengths for specific differences, advancing understanding of their combinatorial structure.

Findings

Maximum progression length in Rudin–Shapiro sequence for difference d ≥ 3 is (d+3)/2.

Exact maximal lengths are computed for differences of the form 2^k-1 and 2^k+1.

Provides upper bounds for progressions in general pattern-based binary sequences.

Abstract

Let $e_v(n)$ denote the number of occurrences of a fixed pattern $v$ in the binary expansion of $n \in \mathbb{N}$. In this paper we study monochromatic arithmetic progressions in the class of binary words $(e_v(n) \bmod{2})_{n \geq 0}$, which includes the famous Thue--Morse word $\mathbf{t}$ and Rudin--Shapiro word $\mathbf{r}$. We prove that the length of a monochromatic arithmetic progression of difference $d \geq 3$ starting at $0$ in $\mathbf{r}$ is at most $(d+3)/2$, with equality for infinitely many $d$. Moreover, we compute the maximal length of a monochromatic arithmetic progression in $\mathbf{r}$ of difference $2^k-1$ and $2^k+1$. 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.