On arithmetic progressions in non-periodic self-affine tilings

Abstract

We study the repetition of patches in self-affine tilings in R^d. In particular, we study the existence and non-existence of arithmetic progressions. We first show that an arithmetic condition of the expansion map for a self-affine tiling implies the non-existence of certain one-dimensional arithmetic progressions. Next, we show that the existence of full-rank infinite arithmetic progressions, pure discrete dynamical spectrum, and limit periodicity are all equivalent for a certain class of self-affine tilings. We finish by giving a complete picture for the existence/non-existence of full-rank infinite arithmetic progressions in the self-similar tilings in R^d.

38 pages, 6 figures

References in corpus (4)

• Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems
• Pure Point Diffraction and Mean, Besicovitch and Weyl Almost Periodicity
• Overlap coincidence to strong coincidence in substitution tiling dynamics
• On arithmetic progressions in model sets

Cited by in corpus (2)

• On long arithmetic progressions in binary Morse-like words
• Monochromatic arithmetic progressions in the Fibonacci, Thue-Morse, and Rudin-Shapiro words