Central limit theorems for simultaneous Diophantine approximations

TL;DR

This paper establishes central limit theorems for the distribution of integer values of linear forms with random coefficients, specifically analyzing the frequency of simultaneous hits into shrinking targets, revealing critical exponents in Diophantine approximation.

Contribution

It provides the first quenched and annealed CLTs for simultaneous Diophantine approximations with random coefficients, identifying the critical exponent for infinite hits.

Findings

Proves quenched and annealed CLTs for the distribution of hits.

Identifies the critical exponent for infinite hits as r/d.

Connects Diophantine approximation with probabilistic limit theorems.

Abstract

We study the distribution modulo $1$ of the values taken on the integers of $r$ linear forms in $d$ variables with random coefficients. We obtain quenched and annealed central limit theorems for the number of simultaneous hits into shrinking targets of radii $n^{-\frac{r}{d}}$. By the Khintchine-Groshev theorem on Diophantine approximations, $\frac{r}{d}$ is the critical exponent for the infinite number of hits.