On absolute continuity of inhomogeneous and contracting on average self-similar measures

We give a condition for absolute continuity of self-similar measures in arbitrary dimensions. This allows us to construct the first explicit absolutely continuous examples of inhomogeneous self-similar measures in dimension one and two. In fact, for $d\ge 1$ and any given rotations in $O(d)$ acting irreducibly on ${\mathbb{R}}^d$ as well as any distinct translations, all having algebraic coefficients, we construct absolutely continuous self-similar measures with the given rotations and translations. We furthermore strengthen Varjú’s result for Bernoulli convolutions, treat complex Bernoulli convolutions and in dimension $\ge 3$ improve the condition on absolute continuity by Lindenstrauss-Varjú. Moreover, self-similar measures of contracting on average measures are studied, which may include expanding similarities in their support.