Central limit theorems in the geometry of numbers

TL;DR

This paper proves that lattice point counting functions and rational approximations in weighted Diophantine settings follow a Central Limit Theorem, using chaotic dynamics of the Cartan flow on lattice spaces.

Contribution

It establishes the CLT for lattice counting in product-defined domains and for rational approximations in weighted Diophantine approximation, linking number theory and dynamical systems.

Findings

CLT holds for lattice points in product-defined domains

CLT applies to rational approximations in weighted Diophantine approximation

Uses chaotic properties of the Cartan flow on lattice spaces

Abstract

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.