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

TL;DR

This paper extends classical discrepancy bounds and logarithm laws to higher-rank lattices, providing new volume estimates and dynamical results that generalize well-known theorems in the geometry of numbers.

Contribution

It introduces higher-rank analogues of discrepancy bounds and logarithm laws for lattices, combining volume estimates with dynamical systems techniques.

Findings

Established higher-rank discrepancy bounds for lattices.

Proved volume estimates for higher minima of lattices.

Derived logarithm laws using dynamical Borel-Cantelli lemmas.

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.