Self-Similar Lattice Tilings and Subdivision Schemes

Abstract

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 $\cup_{n=1}^\infty ( \sum_{j=0}^{n-1} M^j ({\cal D} - {\cal D})) = {\mathbb Z}^s$. Two approaches are provided, one based on scrambling matrices and the other based on primitive matrices. These will follow from the characterization of subdivision schemes associated with nonnegative masks in terms of finite powers of finite matrices, without computing eigenvalues or spectral radii. Our characterization shows that the convergence of the subdivision scheme with a nonnegative mask depends only on the location of its positive coefficients.