Self-affine tiles in ℝⁿ

A self-affine tile in ℝⁿ is a set T of positive measure with A(T) = ∪ _{d ∈ script D} (T + d), where A is an expanding n × n real matrix with |det(A)| = m an integer, and script D = {d, d₂, ..., d_m} ⊆ ℝⁿ is a set of m digits. It is known that self-affine tiles always give tilings of ℝn by translation. This paper extends known characterizations of digit sets script D yielding self-affine tiles. It proves several results about the structure of tilings of ℝⁿ possible using such tiles, and gives examples showing the possible relations between self-replicating tilings and general tilings, which clarify results of Kenyon on self-replicating tilings.