Central Limit Theorems for Diophantine approximants

TL;DR

This paper proves a Central Limit Theorem for counting functions related to solutions of linear inequalities in Diophantine approximation, revealing their random-like behavior through advanced lattice correlation analysis.

Contribution

It introduces a new method analyzing higher-order correlations and cumulants of Siegel transforms to establish probabilistic limit laws in Diophantine approximation.

Findings

Established a Central Limit Theorem for counting functions

Developed a novel technique for estimating cumulants of Siegel transforms

Explained the random-like behavior of solution counting functions

Abstract

In this paper we study counting functions representing the number of solutions of systems of linear inequalities which arise in the theory of Diophantine approximation. We develop a method that allows us to explain the random-like behavior that these functions exhibit and prove a Central Limit Theorem for them. Our approach is based on a quantitative study of higher-order correlations for functions defined on the space of lattices and a novel technique for estimating cumulants of Siegel transforms.