Pablo Shmerkin talk at Cambridge

Pablo Shmerkin gave at talk in the Discrete Analysis Seminar at the Mathematical Sciences Institute, Cambridge on Wednesday 6th February.  The title of the talk was “Normal numbers and fractal measures“.  An abstract of the talk follows.  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 and the square root of 2 are normal, but this problem is so far untractable.  The talk describes 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, results of B. Host and E. Lindenstrauss on normality of p x p invariant measures are completed and extended, as well as many other classical normality results.  The work is joint with M. Hochman.