Poisson approximation and Weibull asymptotics in the geometry of numbers

Minkowski's First Theorem and Dirichlet's Approximation Theorem provide upper bounds on certain minima taken over lattice points contained in domains of Euclidean spaces. We study the distribution of such minima and show, under some technical conditions, that they exhibit Weibull asymptotics with respect to different natural measures on the space of unimodular lattices in $\bR^d$. This follows from very general Poisson approximation results for shrinking targets which should be of independent interest. Furthermore, we show in the appendix that the logarithm laws of Kleinbock-Margulis, Khinchin and Gallagher can be deduced from our distributional results.