  We study natural measures on sets of beta-expansions and on slices through
self similar sets. In the setting of beta-expansions, these allow us to better
understand the measure of maximal entropy for the random beta-transformation
and to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of
equidistribution. Each of these applications is relevant to the study of
Bernoulli convolutions. In the fractal setting this allows us to understand how
to disintegrate Hausdorff measure by slicing, leading to conditions under which
almost every slice through a self similar set has positive Hausdorff measure,
generalising long known results about almost everywhere values of the Hausdorff
dimension.
