Central limit theorems for simultaneous Diophantine approximations

Ilya Vinogradov,Dmitry Dolgopyat,Bassam Fayad

doi:10.5802/jep.37

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.

Highlights

Results. — An important problem in Diophantine approximation is the study of the speed of approach to 0 of a possibly inhomogeneous linear form of several variables evaluated at integers points
Diophantine approximation theory classifies the matrices a and vectors x according to how “resonant” they are; i.e., how well the vector ( xj,αj (k))rj=1 approximates 0 := (0, . . . , 0) ∈ Rr as k varies over a large ball in Zd
Plan of the paper. — Using a standard approach of Dani correspondence we deduce our results about Diophantine approximations from appropriate limit theorems for homogeneous flows

Summary

Introduction

— An important problem in Diophantine approximation is the study of the speed of approach to 0 of a possibly inhomogeneous linear form of several variables evaluated at integers points. If a is uniformly distributed in Tdr, (VN,ι(a, x, c) − VN,ι)/ VN,ι converges in distribution to a normal random variable with zero mean and variance one. — Using a standard approach of Dani correspondence (cf [9, 23, 24, 1, 2, 3, 21]) we deduce our results about Diophantine approximations from appropriate limit theorems for homogeneous flows.

Central Limit Theorems on the space of lattices

Diagonal actions on the space of lattices and Diophantine approximations

An abstract Central Limit Theorem

Preliminaries on diagonal actions and Siegel transforms

Rate of equidistribution of unipotent flows and representative partitions