Dirichlet's theorem on diophantine approximation and homogeneous flows

Dmitry Kleinbock,Barak Weiss

doi:10.3934/jmd.2008.2.43

Abstract

Given an $m \times n$ real matrix $Y$, an unbounded set $\mathcal{T}$of parameters $t =( t_1, \ldots,t_{m+n})\in\mathbb{R}_+^{m+n}$ with $\sum_{i = 1}^m t_i =\sum_{j = 1}^{n} t_{m+j} $ and $0$|Y_i\q - p_i| $|q_j| (here $Y_1,\ldots,Y_m$are rows of $Y$).We show that for any $\varepsilon0$ such that forany drifting away from wallsunbounded $\mathcal{T}$, any $\varepsilon Our results extend those of several authors beginning with the work of Davenportand Schmidt done in late 1960s. The proofs rely on a translation of the probleminto a dynamical one regarding the action of a diagonal semigroup onthe space $\SL_{m+n}(\mathbb R)$/$SL_{m+n}(\mathbb Z)$.

Highlights

Let m, n be positive integers, and denote by Mm,n the space of m × n matrices with real entries
Given Y as above and positive ε < 1, we will say that DT can be ε-improved for Y, and write Y ∈ DIε(m, n), or Y ∈ DIε when the dimensionality is clear from the context, if for every sufficiently large t one can find q ∈ Zn {0} and p ∈ Zm with
Note that Y is called singular if Y ∈ DIε for any ε > 0

Summary

Introduction

Let m, n be positive integers, and denote by Mm,n the space of m × n matrices with real entries. In the case m = 1 we show that for a large class of measures μ (introduced in [KLW]) there is ε0 > 0 such that for any unbounded T , any ε < ε0, and for μ-almost every Y , Dirichlet’s Theorem cannot be ε-improved along T .

Objectives

Results

Conclusion