Digit Frequencies and Bernoulli Convolutions

TL;DR

This paper constructs an explicit set with full Bernoulli convolution measure but Hausdorff dimension less than one, providing insights into the singularity of Bernoulli convolutions related to the golden mean.

Contribution

It explicitly constructs a set with full measure and lower Hausdorff dimension for Bernoulli convolutions, advancing understanding of their singularity properties.

Findings

Constructed an explicit set with measure one and Hausdorff dimension less than one.

Discussed potential generalizations to other Bernoulli convolutions.

Provided a new approach to analyze the singularity of 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 $dim_H(A)<1$. We construct such a set $A$ explicitly and discuss how our approach might be generalised to prove the singularity of other Bernoulli convolutions