Ph.D Student Ph.D Thesis Horesh Tal Euclidean and Non-Euclidean Counting Problems Department of Mathematics PROF. Amos Nevo

Abstract

This thesis is concerned with questions of counting and equidistribution of lattice points in domains defined with respect to the Iwasawa coordinates in non-exceptional real rank-one semi-simple Lie groups. The generality of the rank one case indeed includes counting lattice points in hyperbolic spaces, of any dimension and over any field. All the counting results presented here are effective, namely include an estimation for the error term. Consequentially, the results concerning the equidistribution of the Iwasawa components of lattices in the relevant groups are also effective, namely include an estimation for the rate of convergence.

Several classical Diophantine problems can be reduced to counting lattice points in “Iwasawa domains”: when considering integral-point lattices, e.g. SL(2,Z) inside   SL(2,R), the Iwasawa components represent parameters related to primitive integral vectors. Such parameters include the directions of primitive vectors, and the shortest solution to their associated gcd (greatest common divisor) equation. We establish effective equidistribution of these parameters for primitive vectors of dimension 2, and extend these results to imaginary quadratic number fields. Further applications include counting lifts of closed horospheres to hyperbolic manifolds. The counting results are achieved by utilizing a method due to A. Gorodnik and A. Nevo for an effective counting of lattice points in semi-simple Lie groups.