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Alexander Gorodnik

Publications and source records attributed to Alexander Gorodnik.

53 records · Page 3Linked to original sources
arXiv2010

Lifting, restricting and sifting integral points on affine homogeneous varieties

In a previous paper {GN2} an effective solution of the lattice point counting problem in general domains in semisimple S-algebraic groups and affine symmetric varieties was established. The method relies on the mean ergodic theorem for the action of G on G/Gamma, and implies uniformity in counting over families of lattice subgroups admitting a uniform spectral gap. In the present paper we extend some methods developed in {NS} and use them to establish several useful consequences of this property, including : Effective upper bounds on lifting for solutions of congruences in affine homogeneous varieties, effective upper bounds on the number of integral points on general subvarieties of semisimple group varieties, effective lower bounds on the number of almost prime points on symmetric varieties, and effective upper bounds on almost prime solutions of Linnik-type congruence problems in homogeneous varieties.

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arXiv2010

On Arnold's and Kazhdan's equidistribution problems

We consider isometric actions of lattices in semisimple algebraic groups on (possibly non-compact) homogeneous spaces with (possibly infinite) invariant Radon measure. We assume that the action has a dense orbit, and demonstrate two novel and non-classical dynamical phenomena that arise in this context. The first is the existence of a mean ergodic theorem even when the invariant measure is infinite, which implies the existence of an associated limiting distribution, possibly different than the invariant measure. The second is uniform quantitative equidistribution of all orbits in the space, which follows from a quantitative mean ergodic theorem for such actions. In turn, these results imply quantitative ratio ergodic theorems for isometric actions of lattices. This sheds some unexpected light on certain equidistribution problems posed by Arnol'd and also on the equidistribution conjecture for dense subgroups of isometries formulated by Kazhdan. We briefly describe the general problem regarding ergodic theorems for actions of lattices on homogeneous spaces and its solution via the duality principle \cite{GN2}, and give a number of examples to demonstrate our results. Finally, we also prove results on quantitative equidistribution for transitive actions.

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arXiv2010

Diophantine approximation and automorphic spectrum

The present paper establishes qunatitative estimates on the rate of diophantine approximation in homogeneous varieties of semisimple algebraic groups. The estimates established generalize and improve previous ones, and are sharp in a number of cases. We show that the rate of diophantine approximation is controlled by the spectrum of the automorphic representation, and is thus subject to the generalised Ramanujan conjectures.

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arXiv2009

Counting lattice points

For a locally compact second countable group G and a lattice subgroup Gamma, we give an explicit quantitative solution of the lattice point counting problem in general domains in G, provided that i) G has finite upper local dimension, and the domains satisfy a basic regularity condition, ii) the mean ergodic theorem for the action of G on G/Gamma 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. In addition, it holds uniformly over families of lattices satisfying a uniform spectral gap. Applications include 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. The mean ergodic theorems which we establish are based on spectral methods, including the spectral transfer principle and the Kunze-Stein phenomenon. We prove appropriate analogues of both of these results in the set-up of adele groups, and they constitute a necessary step in our proof of quantitative results in counting rational points.

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arXiv2008

Regularity of conjugacies of algebraic actions of Zariski dense groups

Let α_0 be an affine action of a discrete group Γon a compact homogeneous space X and α_1 a smooth action of Γon X which is C^1-close to α_0. We show that under some conditions, every topological conjugacy between α_0 and α_1 is smooth. In particular, our results apply to Zariski dense subgroups of SL_d(Z) acting on the torus T^d and Zariski dense subgroups of a simple noncompact Lie group G acting on a compact homogeneous space X of G with an invariant measure.

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arXiv2008

Khinchin theorem for integral points on quadratic varieties

We prove an analogue the Khinchin theorem for the Diophantine approximation by integer vectors lying on a quadratic variety. The proof is based on the study of a dynamical system on a homogeneous space of the orthogonal group. We show that in this system, generic trajectories visit a family of shrinking subsets infinitely often.

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arXiv2008

Integral points on symmetric varieties and Satake compatifications

Let V be an affine symmetric variety defined over Q. We compute the asymptotic distribution of the angular components of the integral points in V. This distribution is described by a family of invariant measures concentrated on the Satake boundary of V. In the course of the proof, we describe the structure of the Satake compactifications for general affine symmetic varieties and compute the asymptotic of the volumes of norm balls.

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arXiv2007

The ergodic theory of lattice subgroups

We prove mean and pointwise ergodic theorems for general families of averages on a semisimple algebraic (or S-algebraic) group G, together with an explicit rate of convergence when the action has a spectral gap. Given any lattice in G, we use the ergodic theorems for G to solve the lattice point counting problem for general domains in G, and prove mean and pointwise ergodic theorems for arbitrary measure-preserving actions of the lattice, together with explicit rates of convergence when a spectral gap is present. We also prove an equidistribution theorem in arbitrary isometric actions of the lattice. For the proof we develop a general method to derive ergodic theorems for actions of a locally compact group G, and of a lattice subgroup Gamma, provided certain natural spectral, geometric and regularity conditions are satisfied by the group G, the lattice Gamma, and the domains where the averages are supported. In particular, we establish the general principle that under these conditions a quantitative mean ergodic theorem for a family of averages gives rise to a quantitative solution of the lattice point counting problem in their supports. We demonstrate the new explicit error terms that we obtain by a variety of examples.

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arXiv2007

Strong wavefront lemma and counting lattice points in sectors

We compute the asymptotics of the number of integral quadratic forms with prescribed orthogonal decompositions and, more generally, the asymptotics of the number of lattice points lying in sectors of affine symmetric spaces. A new key ingredient in this article is the strong wavefront lemma, which shows that the generalized Cartan decomposition associated to a symmetric space is uniformly Lipschitz.

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arXiv2005

Ergodicity and mixing of noncommuting epimorphisms

We study mixing properties of epimorphisms of a compact connected finite-dimensional abelian group $X$. In particular, we show that a set $F$, $|F|>\dim X$, of epimorphisms of $X$ is mixing iff every subset of $F$ of cardinality $(\dim X)+1$ is mixing. We also construct examples of free nonabelian groups of automorphisms of tori which are mixing, but not mixing of order 3, and show that, under some irreducibility assumptions, ergodic groups of automorphisms contain mixing subgroups and free nonabelian mixing subsemigroups.

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arXiv2004

Distribution of lattice orbits on homogeneous varieties

Given a lattice Γin a locally compact group G and a closed subgroup H of G, one has a natural action of Γon the homogeneous space V=H\G. For an increasing family of finite subsets {Γ_T: T>0}, a dense orbit vΓ, v\in V, and compactly supported function ϕon V, we consider the sums S_{ϕ,v}(T)=\sum_{γ\in Γ_T} ϕ(v γ). Understanding the asymptotic behavior of S_{ϕ,v}(T) is a delicate problem which has only been considered for certain very special choices of H, G and {Γ_T}. We develop a general abstract approach to the problem, and apply it to the case when G is a Lie group and either H or G is semisimple. When G is a group of matrices equipped with a norm, we have S_{ϕ,v}(T) \sim \int_{G_T} ϕ(vg) dg, where G_T={g\in G:||g||<T} and Γ_T = G_T \cap Γ. We also show that the asymptotics of S_{ϕ,v}(T) is governed by \int_V ϕdν, where νis an explicit limiting density depending on the choice of v and the norm.

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arXiv2004

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

Let X be a symmetric space of noncompact type and Γa lattice in the isometry group of X. We study the distribution of orbits of Γacting on the symmetric space X and its geometric boundary X(\infty). More precisely, for any y in X and b in X(\infty), we investigate the distribution of the set {(yγ,bγ^{-1}):γ\inΓ} in X\times X(\infty). It is proved, in particular, that the orbits of Γin the Furstenberg boundary are equidistributed, and that the orbits of Γin X are equidistributed in ``sectors'' defined with respect to a Cartan decomposition. We also discuss an application to the Patterson-Sullivan theory. Our main tools are the strong wavefront lemma and the equidistribution of solvable flows on homogeneous spaces.

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arXiv2003

Oppenheim conjecture for pairs consisting of a linear form and a quadratic form

Let Q be a nondegenerate quadratic form, and L is a nonzero linear form of dimension d>3. As a generalization of the Oppenheim conjecture, we prove that the set {(Q(x),L(x)):x\in Z^d} is dense in R^2 provided that Q and L satisfy some natural conditions. The proof uses dynamics on homogeneous spaces of Lie groups.

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arXiv2003

Lattice action on the boundary of SL(n,R)

Let Γbe a lattice in G=SL(n,R) and X=G/S a homogeneous space of G, where S is a closed subgroup of G which contains a real algebraic subgroup H such that G/H is compact. We establish uniform distribution of orbits of Γin X analogous to the classical equidistribution on torus. To obtain this result, we first prove an ergodic theorem along balls in the connected component of Borel subgroup of G.

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arXiv2003

Uniform distribution of orbits of lattices on spaces of frames

We study distribution of orbits of a lattice Γ<SL(n,R) in the the space V_{n,l} of l-frames in R^n (1\le l\le n-1). Examples of dense Γ-orbits are known from the work of Dani, Raghavan, and Veech. We show that dense orbits of Γare uniformly distributed in V_{n,l} with respect to an explicitly described measure. We also establish analogous result for lattices in Sp(n,R) that act on the space of isotropic n-frames.

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arXiv2003

On Oppenheim-type conjecture for systems of quadratic forms

Let Q_i, i=1,...,t, be real nondegenerate indefinite quadratic forms in d variables. We investigate under what conditions the closure of the set {(Q_1(x),...,Q_t(x)): x\in Z^d-{0}} contains (0,..,0). As a corollary, we deduce several results on the magnitude of the set Δof g\in GL(d,R) such that the closure of the set {(Q_1(gx),...,Q_t(gx)): x\in Z^d-{0}} contains (0,...,0). Special cases are described when depending on the mutual position of the hypersurfaces {Q_i=0}, i=1,...,t, the set Δhas full Haar measure or measure zero and Hausdorff dimension d^2-(d-2)/2.

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