On arithmetic progressions in non-periodic self-affine tilings

Abstract

Abstract We study the repetition of patches in self-affine tilings in ${\mathbb {R}}^d$ . 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 or non-existence of full-rank infinite arithmetic progressions in the self-similar tilings in ${\mathbb {R}}^d$ .