Monochromatic arithmetic progressions in automatic sequences with group structure

TL;DR

This paper investigates the growth of monochromatic arithmetic progressions in automatic sequences with group structure, providing bounds and asymptotic behavior for such progressions in sequences like Thue--Morse and Rudin--Shapiro.

Contribution

It establishes polynomial growth rates for monochromatic arithmetic progressions in certain automatic sequences and links these bounds to associated finite groups.

Findings

Existence of subsequences with polynomial growth of progression lengths

Explicit bounds derived from associated finite groups

Application to van der Waerden-type numbers

Abstract

We determine asymptotic growth rates for lengths of monochromatic arithmetic progressions in certain automatic sequences. In particular, we look at (one-sided) fixed points of aperiodic, primitive, bijective substitutions and spin substitutions, which are generalisations of the Thue--Morse and Rudin--Shapiro substitutions, respectively. For such infinite words, we show that there exists a subsequence $\left\{d_n\right\}$ of differences along which the maximum length $A(d_n)$ of a monochromatic arithmetic progression (with fixed difference $d_n$) grows at least polynomially in $d_n$. Explicit upper and lower bounds for the growth exponent can be derived from a finite group associated to the substitution. As an application, we obtain bounds for a van der Waerden-type number for a class of colourings parametrised by the size of the alphabet and the length of the substitution.