Self-affine tiles in ℝn

Abstract

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, ..., dm} ⊆ 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. © 1996 Academic Press, Inc.