## On a lower bound for the dimension of Bernoulli convolutions

#### Nikita Sidorov (University of Manchester)

Let $$\beta\in(1,2)$$ and let $$H_\beta$$ denote Garsia's entropy for the Bernoulli convolution $$\mu_\beta$$ associated with $$\beta$$. In my talk I will give a sketch of the proof that $$H_\beta>0.82$$ for all $$\beta \in (1, 2)$$ which is based on a connection between Bernoulli convolutions and $$\beta$$-expansions. Combined with recent results by Hochman and Breuillard-Varjú, this yields $$\dim (\mu_\beta)\ge 0.82$$ for all $$\beta$$.
In addition, I will show that if an algebraic $$\beta$$ is such that $$[\mathbb{Q}(\beta): \mathbb{Q}(\beta^k)] = k$$ for some $$k \geq 2$$ then $$\dim(\mu_\beta)=1$$. Such is, for instance, any root of a Pisot number which is not a Pisot number itself.