Self-Affine Scaling Sets in R2

There exist results on the connection between the theory of wavelets and the theory of integral self-affine tiles and in particular, on the construction of wavelet bases using integral self-affine tiles. However, there are many non-integral self-affine tiles which can also yield wavelet basis. In this work, we give a complete characterization of all one and two dimensional A-dilation scaling sets K such that K is a self-affine tile satisfying BK = (K + d(1))boolean OR(K + d(2)) for some d(1), d(2) is an element of R-2, where A is a 2 x 2 integral expansive matrix with vertical bar det A vertical bar = 2 and B = A(t).