Limit laws in the lattice counting problem. The case of ellipses.

Abstract: Let E be an ellipse centered around 0. We are interested in the asymptotic distribution of the error of the number of unimodular lattice points that fall into tE when the lattice is random and when t goes to infinity. Building on previous works by Bleher and by Fayad and Dolgopyat, we show that the error term, when normalized by the square root of t, converges in distribution towards an explicit distribution. For this, we first use harmonic analysis to reduce the study of the normalized error to the study of a Siegel transform that depends on t. Then, and this is the key part of our proof, we show that, when t goes to infinity, this last Siegel transform behaves in distribution as, what we call, a modified Siegel transform with random weights. Such objects often appear in average counting problems. Finally, we show that this last quantity converges almost surely, and we study the existence of the moments of its law. This work was supervised by Bassam Fayad.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar