  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.
