  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.
