Higher-rank pointwise discrepancy bounds and logarithm laws for generic lattices

Abstract

We prove a higher-rank analogue of a well-known result of W. M. Schmidt concerning almost everywhere pointwise discrepancy bounds for lattices in Euclidean space (see Theorem 1 [Trans. Amer. Math. Soc. 95 (1960), 516-529]). We also establish volume estimates pertaining to higher minima of lattices and then use the work of Kleinbock-Margulis and Kelmer-Yu to prove dynamical Borel-Cantelli lemmata and logarithm laws for higher minima and various related functions.