Central Limit Theorems in Multiplicative Diophantine Approximation

TL;DR

This work proves a central limit theorem for a multiplicative Diophantine counting problem by embedding the problem in homogeneous dynamics on the space of unimodular lattices and analyzing Siegel transforms. A novel decorrelation estimate for translated homogeneous measures, combined with sharp L^p bounds for Siegel transforms on subspaces, enables precise control of variance and all higher cumulants. The variance is expressed via volumes of intersections of scaled domains, and the authors carefully balance smoothing, tessellation, and truncation to obtain Gaussian limit behavior (with a well-defined positive variance) for a broad range of parameters, including a Borel–Cantelli step for small $a$. The methods generalize earlier single-parameter results to a multiplicative, high-dimensional setting, yielding a robust CLT for lattice-point counting in multiplicative Diophantine contexts.

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.

Theorems & Definitions (48)

• Theorem 1.1
• Theorem 1.2
• Theorem 4.1: BFG2,Theorem 1.4
• Theorem 4.2: BFG2, Theorem 6.1
• Theorem 5.1: Siegel sie
• Theorem 5.2: Rogers rog
• Corollary 5.3
• Lemma 5.4: W. Schmidt Schmidt68
• Lemma 5.5
• Proposition 5.6