The Boltzmann-Grad limit of the periodic Lorentz gas and the distribution of visible lattice points

The periodic Lorentz gas describes a particle moving in a periodic array of spherical scatterers, and is one of the fundamental mathematical models for chaotic diffusion in a periodic set-up. In this lecture (aimed at a general mathematical audience) I describe the recent solution of a problem posed by Y. Sinai in the early 1980s, on the nature of the diffusion when the scatterers are very small. The problem is closely related to some basic questions in number theory, in particular the distribution of lattice points visible from a given position, cf. Polya's 1918 paper "[...] ueber die Sichtweite im Walde" (Polya's orchard problem). The key technology in our approach is measure rigidity, a branch of ergodic theory that has proved valuable in recent solutions of other problems in number theory and mathematical physics, such as the value distribution of quadratic forms at integers, quantum unique ergodicity and questions of diophantine approximation.

(This lecture is based on joint work with A. Strombergsson, Uppsala.)