Lattice-tiling properties of integral self-affine functions Original Research Article

Let A be a d × d expanding integer matrix and View the MathML source be absolutely summable and satisfy View the MathML source. A function View the MathML source is called an integralself-affine function for the pair (A, ϱ) if it satisfies the functional equation View the MathML source, a.e. (x). We prove that for such a function there is always a sublattice Λ of Zd such that ƒ tiles Rd with Λ with weight View the MathML source. That is View the MathML source, a.e. (x). The lattice View the MathML source is the smallest A-invariant sublattice of Zd 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.