Integral self-affine tiles in ℝ n part II: Lattice tilings

Abstract

Let A be an expanding n × n integer matrix with |det (A)| = m. A standard digit set \({\cal D}\) for A is any complete set of coset representatives for \({\Bbb Z}^n / A ( {\Bbb Z}^n ).\) Associated to a given \({\cal D}\) is a set \(T( A , {\cal D} ),\) which is the attractor of an affine iterated function system, satisfying \(T = \cup_{d\in {\cal D}} (T + d).\) It is known that \(T( A , {\cal D} )\) tiles \({\Bbb R}^n\) by some subset of \({\Bbb Z}^n.\)This paper proves that every standard digit set \({\cal D}\) gives a set \(T( A , {\cal D} )\) that tiles \({\Bbb R}^n\) with a lattice tiling.