Bernoulli convolutions for algebraic parameters

Special Analysis Seminar
Topic:Bernoulli convolutions for algebraic parameters
Speaker:Péter Varjú
Affiliation:University of Cambridge
Date:Friday, May 8
Time/Room:3:00pm - 4:00pm/S-101

The Bernoulli convolution with parameter $\lambda$ is the law of the random variable: $\sum X_i \lambda^i$, where $X_i$ are independent unbiased $+1/-1$ valued random variables. If $\lambda < 1/2$, then the Bernoulli convolution is singular and is supported on a Cantor set. If $1 > \lambda > 1/2$, the question whether the Bernoulli convolution is singular or a.c. is a very interesting open problem. Recent papers of Hochman and Shmerkin prove that the set of $\lambda$'s such that the measure is singular is of Hausdorff dimension 0. I will discuss the problem for parameters $\lambda$ that are algebraic. Work in progress, joint with Emmanuel Breuillard.