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

Volume 205 / 2022

Seungki Kim, Mishel Skenderi Acta Arithmetica 205 (2022), 227-249 MSC: Primary 11H06; Secondary 11H60, 37A10, 37A17. DOI: 10.4064/aa220325-17-8 Published online: 22 September 2022


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.


  • Seungki KimDepartment of Mathematical Sciences
    University of Cincinnati
    4199 French Hall West
    2815 Commons Way
    Cincinnati, OH 45221-0025, USA
  • Mishel SkenderiDepartment of Mathematics
    The University of Utah
    155 South 1400 East JWB 233
    Salt Lake City, UT 84112-0090, USA

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