Self-Similar Lattice Tilings and Subdivision Schemes

Let $M \in {\mathbb Z}^{s\times s}$ be a dilation matrix and let ${\cal D} \subset {\mathbb Z}^s$ be a complete set of representatives of distinct cosets of ${\mathbb Z}^s/ M{\mathbb Z}^s$. The self-similar tiling associated with M and ${\cal D}$ is the subset of ${\mathbb R}^s$ given by $T(M, {\cal D})=\{ \sum_{j=1}^\infty M^{-j} \alpha_j: \alpha_j \in {\cal D} \}$. The purpose of this paper is to characterize self-similar lattice tilings, i.e., tilings $T(M, {\cal D})$ which have Lebesgue measure one. In particular, it is shown that $T(M, {\cal D})$ is a lattice tiling if and only if there is no nonempty finite set $\Lambda \subset {\mathbb Z}^s \setminus ({\cal D} - {\cal D})$ such that $M^{-1} (({\cal D} - {\cal D}) + \Lambda) \cap {\mathbb Z}^s \subset \Lambda$. This set $\Lambda$ can be restricted to be contained in a finite set K depending only on M and ${\cal D}$. We also give a new proof for the fact that $T(M, {\cal D})$ is a lattice tiling if and only if $\cup_{n=1}^\infty ( \sum_{j=0}^{n-1} M^...