On the exceptional set for absolute continuity of Bernoulli convolutions

TL;DR

This paper proves that the set of parameters for Bernoulli convolutions leading to singular measures has zero Hausdorff dimension, improving previous results and applying to self-similar measures.

Contribution

It establishes that the exceptional set for absolute continuity of Bernoulli convolutions has zero Hausdorff dimension, independent of bias, using new and existing dimension and Fourier decay results.

Findings

Exceptional set has zero Hausdorff dimension.

Results apply to biased Bernoulli convolutions.

Extends to convolutions of self-similar measures.

Abstract

We prove that the set of exceptional $\lambda\in (1/2,1)$ such that the associated Bernoulli convolution is singular has zero Hausdorff dimension, and likewise for biased Bernoulli convolutions, with the exceptional set independent of the bias. This improves previous results by Erd\"os, Kahane, Solomyak, Peres and Schlag, and Hochman. A theorem of this kind is also obtained for convolutions of homogeneous self-similar measures. The proofs are very short, and rely on old and new results on the dimensions of self-similar measures and their convolutions, and the decay of their Fourier transform.