Integral self-affine tiles in ℝn I. Standard and nonstandard digit sets

Abstract

We investigate the measure and tiling properties of integral self-affine tiles, which are sets of positive Lebesgue measure of the form T(A, script D) = {Σ^∞<inf>j=1</inf> A^-j d<inf>j</inf>: all d<inf>j</inf> ∈ script D}, where A ∈ M<inf>n</inf>(ℤ) is an expanding matrix with |det(A)| = m, and script D ⊆ ℤⁿ is a set of m integer vectors. The set script D is called a digit set, and is called standard if it is a complete set of residues of ℤⁿ/A(ℤⁿ) or arises from one by an integer affine transformation, and nonstandard otherwise. We prove that all sets T(A, script D) have integer Lebesgue measure, and study when the measure μ(T(A, script D)) ≠ 0. We give a Fourier-analytic condition for μ(T(A, script D)) ≠ 0. We classify nonstandard digit sets in special cases, and give formulae for the measures of their associated tiles.