Dirichlet’s Theorem states that for a real $mxn$ matrix $A$, $||Aq+p||^m â‰¤ t,$ $ ||q||^n t$ has nontrivial integer solutions for all $t 1.$ Davenport and Schmidt have observed that if $1/t$ is replaced with $c/t, c1,$ almost no $A$ has the property that there exist solutions for all sufficiently large t. Replacing $c/$t with an arbitrary function, it’s natural to ask when precisely does the set of such A drop to a null set. In the case $m=1=n$, we give an answer using dynamics of continued fractions. We then discuss an approach to higher dimensions based on dynamics on the space of lattices. Where this approach reaches an impasse, a similar approach to the analogous inhomogeneous approximation problem will succeed. Joint work with Dmitry Kleinbock.