LATTICE-TILING PROPERTIES OF INTEGRAL SELF-AFFINE FUNCTIONS

Let A be a d x d expanding integer matrix and rho : Z(d) --> C be abso lutely summable and satisfy Sigma(x is an element of Zd) rho(x) = \det A\. A function f is an element of L-1(R-d) is called an integral self -affine function for the pair (A, rho) if it satisfies the functional equation f(A(-1) x) = Sigma(y is an element of Zd) rho(y)f(x - y), a.e . (x). We prove that for such a function there is always a sublattice Lambda of Z(d) such that f tiles R-d with Lambda with weight w = \Z(d) : Lambda\(-1) integral(Rd) f. That is Sigma(lambda is an element of L ambda) f(x - lambda) = w, a.e. (x). The lattice Lambda subset of or eq ual to Z(d) is the smallest A-invariant sublattice of Z(d) that contai ns the support of rho. This generalizes results of Lagarias and Wang [ 1] and others, which were obtained for f and rho which are indicator f unctions of compact sets. The proofs use Fourier Analysis.