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Central limit theorems for generic lattice point counting


Björklund, Michael; Gorodnik, Alexander (2023). Central limit theorems for generic lattice point counting. Selecta Mathematica, 29(1):12.

Abstract

We consider the problem of counting lattice points contained in domains in Rd defined by products of linear forms. For d≥9 we show that the normalized discrepancies in these counting problems satisfy non-degenerate Central Limit Theorems with respect to the unique SLd(R)-invariant probability measure on the space of unimodular lattices in Rd. We also study more refined versions pertaining to “spiraling of approximations”. Our techniques are dynamical in nature and exploit effective exponential mixing of all orders for actions of diagonalizable subgroups on spaces of unimodular lattices.

Abstract

We consider the problem of counting lattice points contained in domains in Rd defined by products of linear forms. For d≥9 we show that the normalized discrepancies in these counting problems satisfy non-degenerate Central Limit Theorems with respect to the unique SLd(R)-invariant probability measure on the space of unimodular lattices in Rd. We also study more refined versions pertaining to “spiraling of approximations”. Our techniques are dynamical in nature and exploit effective exponential mixing of all orders for actions of diagonalizable subgroups on spaces of unimodular lattices.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Scopus Subject Areas:Physical Sciences > General Mathematics
Physical Sciences > General Physics and Astronomy
Uncontrolled Keywords:General Physics and Astronomy, General Mathematics Counting problems, Central limit theorems, Exponential mixing of all orders
Language:English
Date:1 February 2023
Deposited On:31 Dec 2022 07:06
Last Modified:27 Feb 2024 02:53
Publisher:Springer
ISSN:1022-1824
Additional Information:11 (37A25 60F05)
OA Status:Hybrid
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1007/s00029-022-00815-w
Other Identification Number:MR4519647
Project Information:
  • : FunderChalmers University of Technology
  • : Grant ID
  • : Project Title
  • Content: Published Version
  • Language: English
  • Licence: Creative Commons: Attribution 4.0 International (CC BY 4.0)