Lattice-tiling properties of integral self-affine functions

Abstract

Let A be a d × d expanding integer matrix and ϱ : Z d → C be absolutely summable and satisfy Σ x∈ Z d ϱ(x) = | det A| . A function ƒ ∈ L 1( R d) is called an integral self-affine function for the pair ( A, ϱ) if it satisfies the functional equation ƒ(A −1x) = Σ y∈ Z d ϱ(y)ƒ(x - y) , a.e. ( x). We prove that for such a function there is always a sublattice Λ of Z d such that ƒ tiles R d with Λ with weight ω = | Z d : Λ| −1∝ R d ƒ . That is Σ λ∈Λƒ(x - λ) = ω , a.e. ( x). The lattice Λ ⊆ Z d is the smallest A-invariant sublattice of Z d that contains the support of ϱ. This generalizes results of Lagarias and Wang [1] and others, which were obtained for ƒ and ϱ which are indicator functions of compact sets. The proofs use Fourier Analysis.