Central Limit Theorems in Multiplicative Diophantine Approximation

TL;DR

This paper proves that the counting function for solutions to a multiplicative Diophantine approximation problem follows a normal distribution in the limit, using advanced measure correlation techniques.

Contribution

It introduces a novel approach analyzing correlations of measures on homogeneous spaces to establish a central limit theorem in multiplicative Diophantine approximation.

Findings

Counting function converges to a normal distribution

Established new correlation estimates for measures on homogeneous spaces

Applied Siegel transform estimates to Diophantine problems

Abstract

We investigate the number of integer solutions to a multiplicative Diophantine approximation problem and show that the associated counting function converges in distribution to a normal law. Our approach relies on the analysis of correlations of measures on homogeneous spaces, together with estimates for Siegel transforms restricted to subspaces.