On arithmetic progressions in non-periodic tilings

We study the repetition of patches in self-affine tilings. In particular, we study the existence and non-existence of arithmetic progressions. We first show an arithmetic condition of the expansion map implies the non-existence of one-dimensional arithmetic progressions in self-affine tilings. Next, we show that the existence of full-rank infinite arithmetic progression, having pure point dynamical spectrum, and being limit periodic are all equivalent, for a certain class of self-affine tilings. We finish by giving a complete picture for the one-dimensional case.