Abstract

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.