Alexander Gorodnik

Counting lattice points

Abstract For a locally compact second countable group G and a lattice subgroup Γ, we give an explicit quantitative solution of the lattice point counting problem in general domains in G, provided that G has finite upper local dimension, and the domains satisfy a basic regularity condition, the mean ergodic theorem for the action of G on G/Γ holds, with a rate of convergence. The error term we establish matches the best current result for balls in symmetric spaces of simple higher-rank Lie groups, but holds in much greater generality. A significant advantage of the ergodic theoretic approach we use is that the solution to the lattice point counting problem is uniform over families of lattice subgroups provided they admit a uniform spectral gap. In particular, the uniformity property holds for families of finite index subgroups satisfying a quantitative variant of property τ. We discuss a number of applications, including: counting lattice points in general domains in semisimple S-algebraic groups, counting rational points on group varieties with respect to a height function, and quantitative angular (or conical) equidistribution of lattice points in symmetric spaces and in affine symmetric varieties. We note that the mean ergodic theorems which we establish are based on spectral methods, including the spectral transfer principle and the Kunze–Stein phenomenon. We formulate and prove appropriate analogues of both of these results in the set-up of adele groups, and they constitute a necessary ingredient in our proof of quantitative results for counting rational points.

Orbits of discrete subgroups on a symmetric space and the Furstenberg boundary

Let X be a symmetric space of noncompact type, and let Γ be a lattice in the isometry group of X. We study the distribution of orbits of Γ acting on the symmetric space X and its geometric boundary X(∞), generalizing the main equidistribution result of Margulis’s thesis [M, Theorem 6] to higher-rank symmetric spaces. More precisely, for any y ∈ X and b ∈ X(∞), we investigate the distribution of the set {(yγ, bγ^(−1)) : γ ∈ } in X × X(∞). It is proved, in particular, that the orbits of Γ in the Furstenberg boundary are equidistributed and that the orbits of Γ in X are equidistributed in “sectors” defined with respect to a Cartan decomposition. Our main tools are the strong wavefront lemma and the equidistribution of solvable flows on homogeneous spaces, which we obtain using Shah’s result [S, Corollary 1.2] based on Ratner’s measure-classification theorem [R1, Theorem 1].

Integral points on symmetric varieties and Satake compatifications

Let

Strong wavefront lemma and counting lattice points in sectors

V

Uniform distribution of orbits of lattices on spaces of frames

be an affine symmetric variety defined over

Metric Diophantine approximation on homogeneous varieties

\Bbb Q

Quantitative ergodic theorems and their number-theoretic applications

. We compute the asymptotic distribution of the angular components of the integral points in

Khinchin’s theorem for approximation by integral points on quadratic varieties

V

Lattice action on the boundary of SL

. This distribution is described by a family of invariant measures concentrated on the Satake boundary of

(n,\mathbb{R})}

V