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ERA-MS

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.

JMD

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.

JMD

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.

JMD

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.

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