  The dilation equation arises naturally when using a multiresolution analysis
to construct a wavelet basis. We consider solutions in the space of signed
measures, which, after normalization, can be viewed as pseudo-probability
measures. Using probabilistic ideas as well as a notion of binary expansions,
we discuss the existence and absolute continuity of solutions in dimensions one
and two, efficiently recovering previous results in two natural cases. A
central role is played by the development of two variants of the classical
cascade algorithm, which are adapted to the pseudo-probabilistic and elementary
number theoretic context.
