Integral Self-affine Tiles in U I. Standard and Nonstandard Digit Sets

We investigate the measure and tiling properties of integral self-affine tiles, which are sets of positive Lebesgue measure of the form T(A,@) = { £ * x A~'d^: all d}€@}, where AeMn(Z) is an expanding matrix with |det (A)| = m, and Qs ^ 2" is a set of m integer vectors. The set Q> is called a digit set, and is called standard if it is a complete set of residues of Z"/A(Z") or arises from one by an integer affine transformation, and nonstandard otherwise. We prove that all sets T(A, Of) have integer Lebesgue measure, and study when the measure fi(T(A, Si)) ^ 0. We give a Fourier-analytic condition for /z(r(A, Si)) =t 0. We classify nonstandard digit sets in special cases, and give formulae for the measures of their associated tiles.