Integral Self-Affine Tiles in {Bbb R}^n Part II: Lattice Tilings

Let A be an expanding n × n integer matrix with |det (A)| = m. A standard digit set ${cal D}Let A be an expanding n×n integer matrix with |det(A)|=m. Astandard digit set D for A is any complete set of coset representatives for? ⁿ /A(? ⁿ ). Associated to a given D is a setT (A, D), which is the attractor of an affine iterated function system, satisfyingT=∪ _d∈D (T+d). It is known thatT (A, D) tiles? ⁿ by some subset of? ⁿ . This paper proves that every standard digit set D gives a setT (A, D) that tiles? ⁿ with a lattice tiling.