On arithmetic progressions in non-periodic self-affine tilings

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 or non-existence of full-rank infinite arithmetic progressions in the self-similar tilings in R-d.

Automated summary

This study investigates the existence and non-existence of arithmetic progressions in self-affine tilings, examining how the arithmetic condition of the expansion map affects certain one-dimensional arithmetic progressions, and demonstrating the equivalence of full-rank infinite arithmetic progressions, pure discrete dynamical spectrum, and limit-periodicity in certain self-affine tilings. The research provides a comprehensive overview of the presence or absence of full-rank infinite arithmetic progressions in self-similar tilings.