---
_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'
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language:
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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
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...
---
_id: '13091'
abstract:
- lang: eng
  text: We use a function field version of the Hardy–Littlewood circle method to study
    the locus of free rational curves on an arbitrary smooth projective hypersurface
    of sufficiently low degree. On the one hand this allows us to bound the dimension
    of the singular locus of the moduli space of rational curves on such hypersurfaces
    and, on the other hand, it sheds light on Peyre’s reformulation of the Batyrev–Manin
    conjecture in terms of slopes with respect to the tangent bundle.
acknowledgement: The authors are grateful to Paul Nelson, Per Salberger and Jason
  Starr for useful comments. While working on this paper the first author was supported
  by EPRSC grant EP/P026710/1. The research was partially conducted during the period
  the second author served as a Clay Research Fellow, and partially conducted during
  the period he was supported by Dr. Max Rössler, the Walter Haefner Foundation and
  the ETH Zurich Foundation.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Timothy D
  full_name: Browning, Timothy D
  id: 35827D50-F248-11E8-B48F-1D18A9856A87
  last_name: Browning
  orcid: 0000-0002-8314-0177
- first_name: Will
  full_name: Sawin, Will
  last_name: Sawin
citation:
  ama: Browning TD, Sawin W. Free rational curves on low degree hypersurfaces and
    the circle method. <i>Algebra and Number Theory</i>. 2023;17(3):719-748. doi:<a
    href="https://doi.org/10.2140/ant.2023.17.719">10.2140/ant.2023.17.719</a>
  apa: Browning, T. D., &#38; Sawin, W. (2023). Free rational curves on low degree
    hypersurfaces and the circle method. <i>Algebra and Number Theory</i>. Mathematical
    Sciences Publishers. <a href="https://doi.org/10.2140/ant.2023.17.719">https://doi.org/10.2140/ant.2023.17.719</a>
  chicago: Browning, Timothy D, and Will Sawin. “Free Rational Curves on Low Degree
    Hypersurfaces and the Circle Method.” <i>Algebra and Number Theory</i>. Mathematical
    Sciences Publishers, 2023. <a href="https://doi.org/10.2140/ant.2023.17.719">https://doi.org/10.2140/ant.2023.17.719</a>.
  ieee: T. D. Browning and W. Sawin, “Free rational curves on low degree hypersurfaces
    and the circle method,” <i>Algebra and Number Theory</i>, vol. 17, no. 3. Mathematical
    Sciences Publishers, pp. 719–748, 2023.
  ista: Browning TD, Sawin W. 2023. Free rational curves on low degree hypersurfaces
    and the circle method. Algebra and Number Theory. 17(3), 719–748.
  mla: Browning, Timothy D., and Will Sawin. “Free Rational Curves on Low Degree Hypersurfaces
    and the Circle Method.” <i>Algebra and Number Theory</i>, vol. 17, no. 3, Mathematical
    Sciences Publishers, 2023, pp. 719–48, doi:<a href="https://doi.org/10.2140/ant.2023.17.719">10.2140/ant.2023.17.719</a>.
  short: T.D. Browning, W. Sawin, Algebra and Number Theory 17 (2023) 719–748.
date_created: 2023-05-28T22:01:02Z
date_published: 2023-04-12T00:00:00Z
date_updated: 2023-08-01T14:51:57Z
day: '12'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.2140/ant.2023.17.719
external_id:
  arxiv:
  - '1810.06882'
  isi:
  - '000996014700004'
file:
- access_level: open_access
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  date_created: 2023-05-30T08:05:22Z
  date_updated: 2023-05-30T08:05:22Z
  file_id: '13101'
  file_name: 2023_AlgebraNumberTheory_Browning.pdf
  file_size: 1430719
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file_date_updated: 2023-05-30T08:05:22Z
has_accepted_license: '1'
intvolume: '        17'
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issue: '3'
language:
- iso: eng
month: '04'
oa: 1
oa_version: Published Version
page: 719-748
project:
- _id: 26A8D266-B435-11E9-9278-68D0E5697425
  grant_number: EP-P026710-2
  name: Between rational and integral points
publication: Algebra and Number Theory
publication_identifier:
  eissn:
  - 1944-7833
  issn:
  - 1937-0652
publication_status: published
publisher: Mathematical Sciences Publishers
quality_controlled: '1'
scopus_import: '1'
status: public
title: Free rational curves on low degree hypersurfaces and the circle method
tmp:
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  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
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...
---
_id: '9199'
abstract:
- lang: eng
  text: "We associate a certain tensor product lattice to any primitive integer lattice
    and ask about its typical shape. These lattices are related to the tangent bundle
    of Grassmannians and their study is motivated by Peyre's programme on \"freeness\"
    for rational points of bounded height on Fano\r\nvarieties."
acknowledgement: The authors are very grateful to Will Sawin for useful remarks about
  this topic. While working on this paper the first two authors were supported by
  EPSRC grant EP/P026710/1, and the first and last authors by FWF grant P 32428-N35.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Timothy D
  full_name: Browning, Timothy D
  id: 35827D50-F248-11E8-B48F-1D18A9856A87
  last_name: Browning
  orcid: 0000-0002-8314-0177
- first_name: Tal
  full_name: Horesh, Tal
  id: C8B7BF48-8D81-11E9-BCA9-F536E6697425
  last_name: Horesh
- first_name: Florian Alexander
  full_name: Wilsch, Florian Alexander
  id: 560601DA-8D36-11E9-A136-7AC1E5697425
  last_name: Wilsch
  orcid: 0000-0001-7302-8256
citation:
  ama: Browning TD, Horesh T, Wilsch FA. Equidistribution and freeness on Grassmannians.
    <i>Algebra &#38; Number Theory</i>. 2022;16(10):2385-2407. doi:<a href="https://doi.org/10.2140/ant.2022.16.2385">10.2140/ant.2022.16.2385</a>
  apa: Browning, T. D., Horesh, T., &#38; Wilsch, F. A. (2022). Equidistribution and
    freeness on Grassmannians. <i>Algebra &#38; Number Theory</i>. Mathematical Sciences
    Publishers. <a href="https://doi.org/10.2140/ant.2022.16.2385">https://doi.org/10.2140/ant.2022.16.2385</a>
  chicago: Browning, Timothy D, Tal Horesh, and Florian Alexander Wilsch. “Equidistribution
    and Freeness on Grassmannians.” <i>Algebra &#38; Number Theory</i>. Mathematical
    Sciences Publishers, 2022. <a href="https://doi.org/10.2140/ant.2022.16.2385">https://doi.org/10.2140/ant.2022.16.2385</a>.
  ieee: T. D. Browning, T. Horesh, and F. A. Wilsch, “Equidistribution and freeness
    on Grassmannians,” <i>Algebra &#38; Number Theory</i>, vol. 16, no. 10. Mathematical
    Sciences Publishers, pp. 2385–2407, 2022.
  ista: Browning TD, Horesh T, Wilsch FA. 2022. Equidistribution and freeness on Grassmannians.
    Algebra &#38; Number Theory. 16(10), 2385–2407.
  mla: Browning, Timothy D., et al. “Equidistribution and Freeness on Grassmannians.”
    <i>Algebra &#38; Number Theory</i>, vol. 16, no. 10, Mathematical Sciences Publishers,
    2022, pp. 2385–407, doi:<a href="https://doi.org/10.2140/ant.2022.16.2385">10.2140/ant.2022.16.2385</a>.
  short: T.D. Browning, T. Horesh, F.A. Wilsch, Algebra &#38; Number Theory 16 (2022)
    2385–2407.
date_created: 2021-02-25T09:56:57Z
date_published: 2022-12-01T00:00:00Z
date_updated: 2023-08-02T06:46:38Z
day: '01'
department:
- _id: TiBr
doi: 10.2140/ant.2022.16.2385
external_id:
  arxiv:
  - '2102.11552'
  isi:
  - '000961514100004'
intvolume: '        16'
isi: 1
issue: '10'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2102.11552
month: '12'
oa: 1
oa_version: Preprint
page: 2385-2407
project:
- _id: 26A8D266-B435-11E9-9278-68D0E5697425
  grant_number: EP-P026710-2
  name: Between rational and integral points
- _id: 26AEDAB2-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: P32428
  name: New frontiers of the Manin conjecture
publication: Algebra & Number Theory
publication_identifier:
  eissn:
  - 1944-7833
  issn:
  - 1937-0652
publication_status: published
publisher: Mathematical Sciences Publishers
quality_controlled: '1'
scopus_import: '1'
status: public
title: Equidistribution and freeness on Grassmannians
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 16
year: '2022'
...
---
_id: '9260'
abstract:
- lang: eng
  text: We study the density of rational points on a higher-dimensional orbifold (Pn−1,Δ)
    when Δ is a Q-divisor involving hyperplanes. This allows us to address a question
    of Tanimoto about whether the set of rational points on such an orbifold constitutes
    a thin set. Our approach relies on the Hardy–Littlewood circle method to first
    study an asymptotic version of Waring’s problem for mixed powers. In doing so
    we make crucial use of the recent resolution of the main conjecture in Vinogradov’s
    mean value theorem, due to Bourgain–Demeter–Guth and Wooley.
acknowledgement: While working on this paper the authors were both supported by EPSRC
  grant EP/P026710/1, and the second author received additional support from the NWO
  Veni Grant 016.Veni.192.047. Thanks are due to Marta Pieropan, Arne Smeets and Sho
  Tanimoto for useful conversations related to this topic, and to the anonymous referee
  for numerous helpful suggestions.
article_processing_charge: No
article_type: original
author:
- first_name: Timothy D
  full_name: Browning, Timothy D
  id: 35827D50-F248-11E8-B48F-1D18A9856A87
  last_name: Browning
  orcid: 0000-0002-8314-0177
- first_name: Shuntaro
  full_name: Yamagishi, Shuntaro
  last_name: Yamagishi
citation:
  ama: Browning TD, Yamagishi S. Arithmetic of higher-dimensional orbifolds and a
    mixed Waring problem. <i>Mathematische Zeitschrift</i>. 2021;299:1071–1101. doi:<a
    href="https://doi.org/10.1007/s00209-021-02695-w">10.1007/s00209-021-02695-w</a>
  apa: Browning, T. D., &#38; Yamagishi, S. (2021). Arithmetic of higher-dimensional
    orbifolds and a mixed Waring problem. <i>Mathematische Zeitschrift</i>. Springer
    Nature. <a href="https://doi.org/10.1007/s00209-021-02695-w">https://doi.org/10.1007/s00209-021-02695-w</a>
  chicago: Browning, Timothy D, and Shuntaro Yamagishi. “Arithmetic of Higher-Dimensional
    Orbifolds and a Mixed Waring Problem.” <i>Mathematische Zeitschrift</i>. Springer
    Nature, 2021. <a href="https://doi.org/10.1007/s00209-021-02695-w">https://doi.org/10.1007/s00209-021-02695-w</a>.
  ieee: T. D. Browning and S. Yamagishi, “Arithmetic of higher-dimensional orbifolds
    and a mixed Waring problem,” <i>Mathematische Zeitschrift</i>, vol. 299. Springer
    Nature, pp. 1071–1101, 2021.
  ista: Browning TD, Yamagishi S. 2021. Arithmetic of higher-dimensional orbifolds
    and a mixed Waring problem. Mathematische Zeitschrift. 299, 1071–1101.
  mla: Browning, Timothy D., and Shuntaro Yamagishi. “Arithmetic of Higher-Dimensional
    Orbifolds and a Mixed Waring Problem.” <i>Mathematische Zeitschrift</i>, vol.
    299, Springer Nature, 2021, pp. 1071–1101, doi:<a href="https://doi.org/10.1007/s00209-021-02695-w">10.1007/s00209-021-02695-w</a>.
  short: T.D. Browning, S. Yamagishi, Mathematische Zeitschrift 299 (2021) 1071–1101.
date_created: 2021-03-21T23:01:21Z
date_published: 2021-03-05T00:00:00Z
date_updated: 2023-08-07T14:20:00Z
day: '05'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.1007/s00209-021-02695-w
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month: '03'
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oa_version: Published Version
page: 1071–1101
project:
- _id: 26A8D266-B435-11E9-9278-68D0E5697425
  grant_number: EP-P026710-2
  name: Between rational and integral points
publication: Mathematische Zeitschrift
publication_identifier:
  eissn:
  - 1432-1823
  issn:
  - 0025-5874
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Arithmetic of higher-dimensional orbifolds and a mixed Waring problem
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...