Higher-rank pointwise discrepancy bounds and logarithm laws for generic lattices
Volume 205 / 2022
Acta Arithmetica 205 (2022), 227-249
MSC: Primary 11H06; Secondary 11H60, 37A10, 37A17.
DOI: 10.4064/aa220325-17-8
Published online: 22 September 2022
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 in [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.
