---
_id: '14717'
abstract:
- lang: eng
  text: We count primitive lattices of rank d inside Zn as their covolume tends to
    infinity, with respect to certain parameters of such lattices. These parameters
    include, for example, the subspace that a lattice spans, namely its projection
    to the Grassmannian; its homothety class and its equivalence class modulo rescaling
    and rotation, often referred to as a shape. We add to a prior work of Schmidt
    by allowing sets in the spaces of parameters that are general enough to conclude
    the joint equidistribution of these parameters. In addition to the primitive d-lattices
    Λ themselves, we also consider their orthogonal complements in Zn⁠, A1⁠, and show
    that the equidistribution occurs jointly for Λ and A1⁠. Finally, our asymptotic
    formulas for the number of primitive lattices include an explicit bound on the
    error term.
acknowledgement: This work was done when both authors were visiting Institute of Science
  and Technology (IST) Austria. T.H. was being supported by Engineering and Physical
  Sciences Research Council grant EP/P026710/1. Y.K. had a great time there and is
  grateful for the hospitality. The appendix to this paper is largely based on a mini
  course T.H. had given at IST in February 2020.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Tal
  full_name: Horesh, Tal
  id: C8B7BF48-8D81-11E9-BCA9-F536E6697425
  last_name: Horesh
- first_name: Yakov
  full_name: Karasik, Yakov
  last_name: Karasik
citation:
  ama: Horesh T, Karasik Y. Equidistribution of primitive lattices in ℝn. <i>Quarterly
    Journal of Mathematics</i>. 2023;74(4):1253-1294. doi:<a href="https://doi.org/10.1093/qmath/haad008">10.1093/qmath/haad008</a>
  apa: Horesh, T., &#38; Karasik, Y. (2023). Equidistribution of primitive lattices
    in ℝn. <i>Quarterly Journal of Mathematics</i>. Oxford University Press. <a href="https://doi.org/10.1093/qmath/haad008">https://doi.org/10.1093/qmath/haad008</a>
  chicago: Horesh, Tal, and Yakov Karasik. “Equidistribution of Primitive Lattices
    in ℝn.” <i>Quarterly Journal of Mathematics</i>. Oxford University Press, 2023.
    <a href="https://doi.org/10.1093/qmath/haad008">https://doi.org/10.1093/qmath/haad008</a>.
  ieee: T. Horesh and Y. Karasik, “Equidistribution of primitive lattices in ℝn,”
    <i>Quarterly Journal of Mathematics</i>, vol. 74, no. 4. Oxford University Press,
    pp. 1253–1294, 2023.
  ista: Horesh T, Karasik Y. 2023. Equidistribution of primitive lattices in ℝn. Quarterly
    Journal of Mathematics. 74(4), 1253–1294.
  mla: Horesh, Tal, and Yakov Karasik. “Equidistribution of Primitive Lattices in
    ℝn.” <i>Quarterly Journal of Mathematics</i>, vol. 74, no. 4, Oxford University
    Press, 2023, pp. 1253–94, doi:<a href="https://doi.org/10.1093/qmath/haad008">10.1093/qmath/haad008</a>.
  short: T. Horesh, Y. Karasik, Quarterly Journal of Mathematics 74 (2023) 1253–1294.
date_created: 2023-12-31T23:01:03Z
date_published: 2023-12-01T00:00:00Z
date_updated: 2024-01-02T07:39:55Z
day: '01'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.1093/qmath/haad008
external_id:
  arxiv:
  - '2012.04508'
file:
- access_level: open_access
  checksum: bf29baa9eae8500f3374dbcb80712687
  content_type: application/pdf
  creator: dernst
  date_created: 2024-01-02T07:37:09Z
  date_updated: 2024-01-02T07:37:09Z
  file_id: '14720'
  file_name: 2023_QuarterlyJourMath_Horesh.pdf
  file_size: 724748
  relation: main_file
  success: 1
file_date_updated: 2024-01-02T07:37:09Z
has_accepted_license: '1'
intvolume: '        74'
issue: '4'
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
page: 1253-1294
project:
- _id: 26A8D266-B435-11E9-9278-68D0E5697425
  grant_number: EP-P026710-2
  name: Between rational and integral points
publication: Quarterly Journal of Mathematics
publication_identifier:
  eissn:
  - 1464-3847
  issn:
  - 0033-5606
publication_status: published
publisher: Oxford University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: Equidistribution of primitive lattices in ℝn
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 74
year: '2023'
...