On arithmetic progressions in non-periodic self-affine tilings

TL;DR

This paper investigates the conditions under which arithmetic progressions occur in self-affine tilings, revealing deep connections between algebraic properties of the expansion map and the tilings' dynamical spectra.

Contribution

It establishes new links between the algebraic conditions of the expansion map and the existence of arithmetic progressions and spectral properties in self-affine tilings.

Findings

Certain algebraic conditions prevent one-dimensional arithmetic progressions.

Existence of full-rank arithmetic progressions is equivalent to pure discrete spectrum.

Complete characterization of arithmetic progressions in self-similar 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.