Absolute continuity of complex Bernoulli convolutions

TL;DR

This paper proves that complex Bernoulli convolutions are absolutely continuous for most parameters, except for a negligible set, extending previous results to biased cases and other self-similar measures in the complex plane.

Contribution

It establishes absolute continuity of complex Bernoulli convolutions in the supercritical region, including biased cases and broader families, with negligible exceptional sets.

Findings

Absolute continuity holds outside a zero Hausdorff dimension set

Results extend to biased Bernoulli convolutions

Applicable to various parametrized self-similar measures in the complex plane

Abstract

We prove that complex Bernoulli convolutions are absolutely continuous in the supercritical parameter region, outside of an exceptional set of parameters of zero Hausdorff dimension. Similar results are also obtained in the biased case, and for other parametrized families of self-similar sets and measures in the complex plane, extending earlier results.