We present results on the existence of long arithmetic progressions in the Thue-Morse word and in a class of generalised Thue-Morse words. Our arguments are inspired by van der Waerden's proof for the existence of arbitrary long monochromatic arithmetic progressions in any finite colouring of the (positive) integers.

翻译：我们用Thue-Morse字和一般的Thue-Morse字来提出长期算术进展的结果。 我们的论点来自van der Waerden证明在(正)整数的任何限定颜色中存在任意的长长单色算术进步的证据。