Digit frequencies and Bernoulli convolutions

It is well known that when ββ is a Pisot number, the corresponding Bernoulli convolution νβνβ has Hausdorff dimension less than 11, i.e. that there exists a set AβAβ with νβ(Aβ)=1νβ(Aβ)=1 and dimH(Aβ)<1dimH(Aβ)<1. We show explicitly how to construct for each Pisot number ββ such a set AβAβ.