A Bernoulli convolution is the measure defined by the random series $$\sum_{n=1}^{\infty} \pm\lambda^n$$ where the signs are chosen with probability $p$ and $1-p$ and $1/2<\lambda<1$. I will begin by surveying some known results about these measures in particular the issue of when they are singular or absolutely continuous. Most of the known results here are either for typical values of $\lambda$ or for specific algebraic integers.

In the second part of the talk I will describe some recent joint work with Pablo Shmerkin and Boris Solomyak. In this work we consider more local properties of Bernoulli convolutions in particular the local dimension.
For absolutely continuous Bernoulli convolutions the local dimension is almost surely 1 the aim of our work is to look at whether other values can be taken. This is based on work in
http://arxiv.org/abs/1011.1938 and some more recent progress.