Digit Frequencies and Bernoulli Convolutions

It is well known that the Bernoulli convolution $\nu_{\beta}$ associated to the golden mean has Hausdorff dimension less than 1, i.e. that there exists a set $A$ with $\nu_{\beta}(A)=1$ and $dim_H(A)<1$. We construct such a set $A$ explicitly and discuss how our approach might be generalised to prove the singularity of other Bernoulli convolutions