On a lower bound for the dimension of Bernoulli convolutions

Nikita Sidorov (University of Manchester)

Frank Adams 1,

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.

This talk is based on my recent paper with Kevin Hare (Waterloo).

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