Alexander Gorodnik

Integral points on symmetric varieties and Satake compatifications (2009)

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...

Integral points on symmetric varieties and Satake compatifications (2009)

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...

Integral points on symmetric varieties and Satake compatifications (2009)

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...

Alexander Gorodnik, Hee Oh, Nimish Shah

American Journal of Mathematics - Volume 131, Number 1, February 2009

Khinchin theorem for integral points on quadratic varieties (2008)

Gorodnik, Alexander, Shah, Nimish A.

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...

Regularity of conjugacies of algebraic actions of Zariski dense groups (2008)

Gorodnik, Alexander, Hitchman, Theron, Spatzier, Ralf

Let \alpha_0 be an affine action of a discrete group \Gamma on a compact homogeneous space X and \alpha_1 a smooth action of \Gamma on X which is C^1-close to \alpha_0. We show that under some...

Ergodicity and mixing of non-commuting epimorphisms (2007)

Strong wavefront lemma and counting lattice points in sectors (2007)

We study mixing properties of epimorphisms of a compact connected finite-dimensional abelian group X. In particular, we show that a set F, with |F| > dim X, of epimorphisms of X is mixing if and only...

Ergodicity and mixing of non-commuting epimorphisms (2007)

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...

Integral points on symmetric varieties and Satake compatifications (2006)

We study mixing properties of epimorphisms of a compact connected finite-dimensional abelian group X. In particular, we show that a set F, with |F| > dim X, of epimorphisms of X is mixing if and only...

The ergodic theory of lattice subgroups (2006)

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...

Gorodnik, Alexander, Nevo, Amos

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...

Ergodicity and mixing of noncommuting epimorphisms (2005)

Weakly mixing group actions: a brief survey and an example (2005)

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...

Proximality and equidistribution on the Furstenberg boundary (2005)

The first part of the paper contains a survey of weakly mixing group actions of general group, and the second part discusses a special example of a weakly mixing action -- the SL(2,Z)-action on the...

Gorodnik, Alexander, Maucourant, Francois

Distribution of lattice orbits on homogeneous varieties (2004)

Gorodnik, Alexander, Weiss, Barak

Given a lattice \Gamma in a locally compact group G and a closed subgroup H of G, one has a natural action of \Gamma on the homogeneous space V=H\G. For an increasing family of finite subsets...

Orbits of discrete subgroups on a symmetric space and the Furstenberg boundary (2004)

Gorodnik, Alexander, Oh, Hee

Let X be a symmetric space of noncompact type and \Gamma a lattice in the isometry group of X. We study the distribution of orbits of \Gamma acting on the symmetric space X and its geometric boundary...

Uniform distribution of orbits of lattices on spaces of frames (2004)

On Oppenheim-type conjecture for systems of quadratic forms (2003)

We study distribution of orbits of a lattice $\Gamma\subseteq\SL(n,\mathbb{R})$ in the the space $\mathcal{V}_{n,l}$ of $l$-frames in $\mathbb{R}^n$ ($1\le l\le n-1$). Examples of dense...

Oppenheim conjecture for pairs consisting of a linear form and a quadratic form (2003)

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)....

Lattice action on the boundary of SL(n,R) (2003)

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...

Uniform distribution of orbits of lattices on spaces of frames (2003)

Let \Gamma be 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...