Normal numbers and fractal measures

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• Pablo Shmerkin (Surrey)
• Wednesday 06 February 2013, 16:00-17:00
• MR11, CMS.

If you have a question about this talk, please contact Bateman.

It is known from E. Borel that almost all real numbers are normal to all integer bases. On the other hand, it is conjectured that natural constants such as $\pi$, $e$ or $\sqrt{2}$ are normal, but this problem is so far untractable. In the talk I will describe a new dynamical approach to an intermediate problem: are ``natural’’ fractal measures supported on numbers normal to a given base? Our results are formulated in terms of an auxiliary flow that reflects the structure of the measure as one zooms in towards a point.

Unlike classical methods based on the Fourier transform, our approach allows to establish normality in some non-integer bases and is robust under smooth perturbations of the measure. As applications, we complete and extend results of B. Host and E. Lindenstrauss on normality of $\times p$ invariant measures, and many other classical normality results. This is a joint work with M. Hochman.

This talk is part of the Discrete Analysis Seminar series.

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