Ph.D Student Ph.D Thesis Bersudsky Michael Equidistribution of Some Expanding Sets in Homogeneous Spaces Department of Mathematics ASSOCIATE PROF. Uri Shapira

Abstract

Motivated by many applications in number theory, a main line of research in homogeneous dynamics studies expanding subsets of a homogeneous space with the goal of describing their asymptotic behavior. In this thesis, we present two novel results in that direction.

In the first part of the thesis we prove the equidistribution of a sequence of expanding periodic orbits in an S-arithmetic space, which is used to compute the statistics of SLd(Z) matrices lying on level sets of an integral polynomial defined on SLd(R); a result that is a variant of the well-known theorem proved by Linnik about the equidistribution of radially projected integral vectors from a large sphere into the unit sphere. The main motivation behind this project is a generalization of the work of Aka, Einsiedler and Shapira in various directions. For example, we compute the joint distribution of the residue classes modulo q and the properly normalized orthogonal lattices of primitive integral vectors lying on the level set -x12-x22-x3242=N as N grows to infinity, where the normalized orthogonal lattices sit in a submanifold of the moduli space of rank-3 discrete subgroups of R4.

In the second part of the thesis, we study the statistics modulo one of sparse points on expanding analytic curves. It is known that the image in R2/Z2 of a circle of radius r in the plane becomes equidistributed as r grows to infinity. We consider the following sparse version of this phenomenon. Starting from a sequence of radii {rn}n>1 which diverges to infinity and an angle w, we consider the projection to R2/Z2 of the nth roots of unity rotated by angle w and dilated by a factor of rn. We prove that if rn grows polynomially in n, then the image of these sparse collections becomes equidistributed, and moreover, if rn grows arbitrarily fast, then we show that equidistribution holds for almost all w. Interestingly, we found that for any angle w there is a sequence of radii growing to infinity faster than any polynomial for which equidistribution fails dramatically. In greater generality, we prove this type of results for dilations of varying analytic curves in Rd. A novel component of the proof is the use of the theory of o-minimal structures to control exponential sums.