Absolute continuity of self-similar measures, their projections and convolutions (Q2790651)

The paper deals with the one-parameter families of measures on the line. Let \(E\) be set of values of the parameter such that the corresponding measure is not absolutely continuous. The authors establish conditions under which the set \(E\) has Hausdorff dimension less than \(1.\) Their result is valid for: {\parindent=6mm \begin{itemize} \item[-] fairly general parametrized families of homogeneous self-similar measures on the line (which include Bernoulli convolutions as a special case); \item [-] projections of homogeneous self-similar measures on the plane (possibly containing a scaled irrational rotation); \item [-] convolutions of scaled homogeneous self-similar measures on the line (the parameter comes in the scaling). NEWLINENEWLINE\end{itemize}} These results have important consequences, in particular, a strong version of Marstrand's projection theorem for planar self-similar sets.