On long arithmetic progressions in binary Morse-like words

Abstract

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.

23 pages; comments on binary/non-binary bijective case clarified and result for the binary case stated explicitly in proposition 34

References in corpus (2)

• Normality of the Thue--Morse sequence along Piatetski-Shapiro sequences
• On arithmetic progressions in non-periodic self-affine tilings

Cited by in corpus (1)

• Monochromatic arithmetic progressions in the Fibonacci, Thue-Morse, and Rudin-Shapiro words