Counting in generic lattices and higher rank actions

TL;DR

This paper proves that lattice point counting in certain domains exhibits normal distribution behavior under specific conditions, using advanced dynamical systems techniques related to higher-rank abelian group actions.

Contribution

It establishes non-degenerate Central Limit Theorems for lattice point counts in high-dimensional settings and introduces refined analysis of approximation spiraling.

Findings

Normalized discrepancies follow CLT for dimensions d ≥ 9

Effective exponential mixing is key to the proofs

Refined results on spiraling of approximations

Abstract

We consider the problem of counting lattice points contained in domains in $\mathbb{R}^d$ defined by products of linear forms and we show that the normalized discrepancies in these counting problems satisfy non-degenerate Central Limit Theorems, provided that $d \geq 9$. We also study more refined versions pertaining to "spiraling of approximations". Our techniques are dynamical in nature and exploit effective exponential mixing of all orders for actions of higher-rank abelian groups on the space of unimodular lattices.