Monochromatic arithmetic progressions in automatic sequences with group structure

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 $A\left(d_n\right)$ 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.