Digit Frequencies and Bernoulli Convolutions

Abstract

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 $di{m}_H(A)<1$. We construct such a set $A$ explicitly and discuss how our approach might be generalised to prove the singularity of other Bernoulli convolutions