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

TL;DR

This paper demonstrates that in many parameterized families of self-similar measures, the set of parameters where the measure is not absolutely continuous is very small, and it establishes regularity of the density outside this set, with applications to Bernoulli convolutions and Marstrand's projection theorem.

Contribution

It shows that the exceptional set of parameters for non-absolutely continuous measures has co-dimension at least one, and proves new results on measure dimensions and convolutions.

Findings

The set of parameters with non-absolutely continuous measures has co-dimension at least one.

Regularity of the density is established outside the small exceptional set.

A strong version of Marstrand's projection theorem for planar self-similar sets is obtained.

Abstract

We show that in many parametrized families of self-similar measures, their projections, and their convolutions, the set of parameters for which the measure fails to be absolutely continuous is very small - of co-dimension at least one in parameter space. This complements an active line of research concerning similar questions for dimension. Moreover, we establish some regularity of the density outside this small exceptional set, which applies in particular to Bernoulli convolutions; along the way, we prove some new results about the dimensions of self-similar measures and the absolute continuity of the convolution of two measures. As a concrete application, we obtain a very strong version of Marstrand's projection theorem for planar self-similar sets.