Variations on Dirichlet's theorem

TL;DR

This paper generalizes Dirichlet's theorem by providing a necessary and sufficient condition for approximation properties involving parameters a and A, analyzing exceptions, and extending results to Diophantine spaces and the case d=1.

Contribution

It introduces a generalized approximation condition with parameters a and A, characterizes exceptions, and extends the analysis to Diophantine spaces and the one-dimensional case.

Findings

Characterizes when the approximation property holds for given parameters.

Shows the set of exceptions is comeager in certain spaces.

Extends results to rational quadratic hypersurfaces and the case d=1.

Abstract

We give a necessary and sufficient condition for the following property of an integer $d\in\mathbb N$ and a pair $(a,A)\in\mathbb R^2$: There exist $\kappa > 0$ and $Q_0\in\mathbb N$ such that for all $\mathbf x\in \mathbb R^d$ and $Q\geq Q_0$, there exists $\mathbf p/q\in\mathbb Q^d$ such that $1\leq q\leq Q$ and $\|\mathbf x - \mathbf p/q\| \leq \kappa q^{-a} Q^{-A}$. This generalizes Dirichlet's theorem, which states that this property holds (with $\kappa = Q_0 = 1$) when $a = 1$ and $A = 1/d$. We also analyze the set of exceptions in those cases where the statement does not hold, showing that they form a comeager set. This is also true if $\mathbb R^d$ is replaced by an appropriate "Diophantine space", such as a nonsingular rational quadratic hypersurface which contains rational points. Finally, in the case $d = 1$ we describe the set of exceptions in terms of classical Diophantine conditions.