Integral Self-Affine Tiles in ℝ n I. Standard and Nonstandard Digit Sets

We investigate the measure and tiling properties of integral self-affine tiles, which are sets of positive Lebesgue measure of the form T(A, 𝒟) = { Σ j = 1 ∞ A−j: all dj ∈ 𝒟}, where A ∈ Mn(Z) is an expanding matrix with |det (A)| = m, and 𝒟 ⊆ Zn is a set of m integer vectors. The set 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, 𝒟) have integer Lebesgue measure, and study when the measure µ(T(A, 𝒟)) ≠ 0. We give a Fourier-analytic condition for µ(T(A, 𝒟)) ≠ 0. We classify nonstandard digit sets in special cases, and give formulae for the measures of their associated tiles.