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.

我们确定某些自动序列中单色算术级数长度的渐近增长率。特别是，我们研究非周期、本原、双射替换和自旋替换的（单边）不动点，它们分别是 Thue-Morse 和 Rudin-Shapiro 替换的推广。对于这样的无限单词，我们证明存在一个子序列$\left\{d_n\right\}$最大长度的差异$A（d_n）$单色算术级数（具有固定差值）$d_n$) 至少以多项式增长$d_n$。增长指数的显式上限和下限可以从与替换相关的有限群中导出。作为一个应用，我们获得一类着色的 van der Waerden 型数字的界限，该颜色由字母表的大小和替换的长度参数化。