Counting orbits of integral points in families of affine homogeneous varieties and diagonal flows
Alexander Gorodnik Frédéric Paulin
Journal of Modern Dynamics 2014, 8(1): 25-59 doi: 10.3934/jmd.2014.8.25
In this paper, we study the distribution of integral points on parametric families of affine homogeneous varieties. By the work of Borel and Harish-Chandra, the set of integral points on each such variety consists of finitely many orbits of arithmetic groups, and we establish an asymptotic formula (on average) for the number of the orbits indexed by their Siegel weights. In particular, we deduce asymptotic formulas for the number of inequivalent integral representations by decomposable forms and by norm forms in division algebras, and for the weighted number of equivalence classes of integral points on sections of quadrics. Our arguments use the exponential mixing property of diagonal flows on homogeneous spaces.
keywords: homogeneous variety counting norm form exponential decay of correlation. Siegel weight diagonalizable flow Integral point mixing decomposable form
Central limit theorems in the geometry of numbers
Michael Björklund Alexander Gorodnik
Electronic Research Announcements 2017, 24(0): 110-122 doi: 10.3934/era.2017.24.012

We investigate in this paper the distribution of the discrepancy of various lattice counting functions. In particular, we prove that the number of lattice points contained in certain domains defined by products of linear forms satisfies a Central Limit Theorem. Furthermore, we show that the Central Limit Theorem holds for the number of rational approximants for weighted Diophantine approximation in $\mathbb{R}^d$. Our arguments exploit chaotic properties of the Cartan flow on the space of lattices.

keywords: Central Limit Theorems Diophantine approximation
Open problems in dynamics and related fields
Alexander Gorodnik
Journal of Modern Dynamics 2007, 1(1): 1-35 doi: 10.3934/jmd.2007.1.1
The paper discusses a number of open questions,which were collected during the AIM workshop “Emerging applications of measure rigidity”. The main emphasis is made on the rigidity problems in the theory of dynamical systems and their connections with Diophantine approximation, arithmetic geometry, and quantum chaos.
keywords: measure rigidity André-Oort conjecture arithmetic groups symbolic coding. equidistribution Diophantine approximation local rigidity global rigidity polygonal billiards Littlewood conjecture quantumchaos divergent orbits
Regularity of conjugacies of algebraic actions of Zariski-dense groups
Alexander Gorodnik Theron Hitchman Ralf Spatzier
Journal of Modern Dynamics 2008, 2(3): 509-540 doi: 10.3934/jmd.2008.2.509
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 conditions, every topological conjugacy between $\alpha_0$ and $\alpha_1$ is smooth. In particular, our results apply to Zariski-dense subgroups of $SL_d(\mathbb{Z})$ acting on the torus $\mathbb{T}^d$ and Zariski-dense subgroups of a simple noncompact Lie group $G$ acting on a compact homogeneous space $X$ of $G$ with an invariant measure.
keywords: Group actions rigidityh. conjugacies

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