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

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