Self-similar lattice tilings and subdivision schemes (Q2719180)

Let \( M \) be an \( s \times s\) integer matrix with all eigenvalues greater than 1 in modulus, and let \( {\mathcal D} \) be a complete set of representatives of distinct cosets of the quotient group \( {\mathbb Z}^s /M {\mathbb Z}^s \). NEWLINENEWLINENEWLINEA self-similar lattice tiling is defined as a subset of \({\mathbb R}^s \) of the form NEWLINE\[NEWLINE T(M,{\mathcal D}) := \Biggl\{ \sum^{\infty}_{j=1} M^{-j} \alpha_j: \alpha_j \in {\mathcal D} \Biggr\} NEWLINE\]NEWLINE with Lebesgue measure one. NEWLINENEWLINENEWLINEThe paper presents a new proof of the fact that \( T(M,{\mathcal D})\) is a self-similar lattice tiling if and only if \(\bigcup_{n=1}^{\infty} \sum^{n-1}_{j=0} M^j ({\mathcal D} - {\mathcal D}) = {\mathbb Z}^s , \) where \( {\mathcal D} - {\mathcal D} := \{ \alpha - \beta: \alpha,\beta \in {\mathcal D}\}\). NEWLINENEWLINENEWLINEFurther, it is shown that \(T(M,{\mathcal D}) \) is a lattice tiling if and only if there is no nonempty finite set \(\Lambda \subset {\mathbb Z}^s ({\mathcal D} -{\mathcal D}) \) such that \( M^{-1} (({\mathcal D} -{\mathcal D})- \Lambda) \cap {\mathbb Z}^s \subset \Lambda \). Here \( \Lambda \) can be restricted to be contained in a finite set \( K \) depending on \( M \) and \( {\mathcal D} \). NEWLINENEWLINENEWLINEThe proofs are based on scrambling matrices and on primitive matrices, which also occur in the characterization of subdivision schemes with nonnegative masks. NEWLINENEWLINENEWLINEThe convergence of such subdivision schemes is shown to depend only on the location of its positive coefficients.