Translational self-affine tiles in $\mathbb{R}^n$

Abstract

Two basic problems in the theory of integral self-affine tiles are to characterize the digit sets that generate translational tiles and to understand whether such tiles admit periodic translational tilings. We give a complete solution to the first problem: for an expanding integer matrix $\mathbf{A}$ and a digit set $\mathcal{D}\subset\mathbb{Z}^n$ with $\#\mathcal{D}=|\det\mathbf{A}|$, the self-affine set $T(\mathbf{A},\mathcal{D})$ is a tile if and only if, for some $N\geq1$, the iterated digit set $\mathcal{D}_{\mathbf{A},N}$ is of skew-product form with respect to $\mathbf{A}^N$. As a consequence, every integral self-affine tile admits a fully periodic self-replicating $q$-fold tiling. Moreover, $q$ divides $|T(\mathbf{A},\mathcal{D})|$. In dimension two we prove more: every integral self-affine tile admits a fully periodic tiling. In particular, the periodic tiling conjecture holds for integral self-affine tiles in $\mathbb{R}^2$.