Integral Self-Affine Tiles in ℝn Part II: Lattice Tilings

Abstract

Let A be an expanding n × n integer matrix with | det(A)| = m. A standard digit set D for A is any complete set of coset representatives for ℤⁿ/A(ℤⁿ). Associated to a given D is a set T(A, D), which is the attractor of an affine iterated function system, satisfying T = U<inf>dεD</inf>(T + d). It is known that T(A, D) tiles ℝⁿ by some subset of ℤⁿ. This paper proves that every standard digit set D gives a set T(A, D) that tiles ℝⁿ with a lattice tiling.