Variations on Dirichlet's theorem

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 $Q\ge Q_0$, there exists $\mathbf{p}/q\in {\mathbb{Q}}^d$ such that $1\le q\le Q$ and $\Vert \mathbf{x}-\mathbf{p}/q\Vert \le \kappa q^{-a}{Q}^{-A}$. 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.