---
_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. Quarterly
Journal of Mathematics. 2023;74(4):1253-1294. doi:10.1093/qmath/haad008
apa: Horesh, T., & Karasik, Y. (2023). Equidistribution of primitive lattices
in ℝn. Quarterly Journal of Mathematics. Oxford University Press. https://doi.org/10.1093/qmath/haad008
chicago: Horesh, Tal, and Yakov Karasik. “Equidistribution of Primitive Lattices
in ℝn.” Quarterly Journal of Mathematics. Oxford University Press, 2023.
https://doi.org/10.1093/qmath/haad008.
ieee: T. Horesh and Y. Karasik, “Equidistribution of primitive lattices in ℝn,”
Quarterly Journal of Mathematics, 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.” Quarterly Journal of Mathematics, vol. 74, no. 4, Oxford University
Press, 2023, pp. 1253–94, doi:10.1093/qmath/haad008.
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:
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month: '12'
oa: 1
oa_version: Published Version
page: 1253-1294
project:
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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'
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title: Equidistribution of primitive lattices in ℝn
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...
---
_id: '14245'
abstract:
- lang: eng
text: We establish effective counting results for lattice points in families of
domains in real, complex and quaternionic hyperbolic spaces of any dimension.
The domains we focus on are defined as product sets with respect to an Iwasawa
decomposition. Several natural diophantine problems can be reduced to counting
lattice points in such domains. These include equidistribution of the ratio of
the length of the shortest solution (x,y) to the gcd equation bx−ay=1 relative
to the length of (a,b), where (a,b) ranges over primitive vectors in a disc whose
radius increases, the natural analog of this problem in imaginary quadratic number
fields, as well as equidistribution of integral solutions to the diophantine equation
defined by an integral Lorentz form in three or more variables. We establish an
effective rate of convergence for these equidistribution problems, depending on
the size of the spectral gap associated with a suitable lattice subgroup in the
isometry group of the relevant hyperbolic space. The main result underlying our
discussion amounts to establishing effective joint equidistribution for the horospherical
component and the radial component in the Iwasawa decomposition of lattice elements.
acknowledgement: The authors thank the referee for important comments which led to
significant improvements is the presentation of several results in the paper. They
also thank Ami Paz for preparing the figures for this paper. Horesh thanks Ami Paz
and Yakov Karasik for helpful discussions. Nevo thanks John Parker and Rene Rühr
for providing some very useful references. Nevo is supported by ISF Grant No. 2095/15.
article_processing_charge: Yes
article_type: original
arxiv: 1
author:
- first_name: Tal
full_name: Horesh, Tal
id: C8B7BF48-8D81-11E9-BCA9-F536E6697425
last_name: Horesh
- first_name: Amos
full_name: Nevo, Amos
last_name: Nevo
citation:
ama: 'Horesh T, Nevo A. Horospherical coordinates of lattice points in hyperbolic
spaces: Effective counting and equidistribution. Pacific Journal of Mathematics.
2023;324(2):265-294. doi:10.2140/pjm.2023.324.265'
apa: 'Horesh, T., & Nevo, A. (2023). Horospherical coordinates of lattice points
in hyperbolic spaces: Effective counting and equidistribution. Pacific Journal
of Mathematics. Mathematical Sciences Publishers. https://doi.org/10.2140/pjm.2023.324.265'
chicago: 'Horesh, Tal, and Amos Nevo. “Horospherical Coordinates of Lattice Points
in Hyperbolic Spaces: Effective Counting and Equidistribution.” Pacific Journal
of Mathematics. Mathematical Sciences Publishers, 2023. https://doi.org/10.2140/pjm.2023.324.265.'
ieee: 'T. Horesh and A. Nevo, “Horospherical coordinates of lattice points in hyperbolic
spaces: Effective counting and equidistribution,” Pacific Journal of Mathematics,
vol. 324, no. 2. Mathematical Sciences Publishers, pp. 265–294, 2023.'
ista: 'Horesh T, Nevo A. 2023. Horospherical coordinates of lattice points in hyperbolic
spaces: Effective counting and equidistribution. Pacific Journal of Mathematics.
324(2), 265–294.'
mla: 'Horesh, Tal, and Amos Nevo. “Horospherical Coordinates of Lattice Points in
Hyperbolic Spaces: Effective Counting and Equidistribution.” Pacific Journal
of Mathematics, vol. 324, no. 2, Mathematical Sciences Publishers, 2023, pp.
265–94, doi:10.2140/pjm.2023.324.265.'
short: T. Horesh, A. Nevo, Pacific Journal of Mathematics 324 (2023) 265–294.
date_created: 2023-08-27T22:01:18Z
date_published: 2023-07-26T00:00:00Z
date_updated: 2023-12-13T12:19:42Z
day: '26'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.2140/pjm.2023.324.265
external_id:
arxiv:
- '1612.08215'
isi:
- '001047690500001'
file:
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checksum: a675b53cfb31fa46be1e879b7e77fe8c
content_type: application/pdf
creator: dernst
date_created: 2023-09-05T07:26:17Z
date_updated: 2023-09-05T07:26:17Z
file_id: '14267'
file_name: 2023_PacificJourMaths_Horesh.pdf
file_size: 654895
relation: main_file
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file_date_updated: 2023-09-05T07:26:17Z
has_accepted_license: '1'
intvolume: ' 324'
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issue: '2'
language:
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month: '07'
oa: 1
oa_version: Published Version
page: 265-294
publication: Pacific Journal of Mathematics
publication_identifier:
eissn:
- 1945-5844
issn:
- 0030-8730
publication_status: published
publisher: Mathematical Sciences Publishers
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'Horospherical coordinates of lattice points in hyperbolic spaces: Effective
counting and equidistribution'
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)
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type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 324
year: '2023'
...
---
_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.
Algebra & Number Theory. 2022;16(10):2385-2407. doi:10.2140/ant.2022.16.2385
apa: Browning, T. D., Horesh, T., & Wilsch, F. A. (2022). Equidistribution and
freeness on Grassmannians. Algebra & Number Theory. Mathematical Sciences
Publishers. https://doi.org/10.2140/ant.2022.16.2385
chicago: Browning, Timothy D, Tal Horesh, and Florian Alexander Wilsch. “Equidistribution
and Freeness on Grassmannians.” Algebra & Number Theory. Mathematical
Sciences Publishers, 2022. https://doi.org/10.2140/ant.2022.16.2385.
ieee: T. D. Browning, T. Horesh, and F. A. Wilsch, “Equidistribution and freeness
on Grassmannians,” Algebra & Number Theory, 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 & Number Theory. 16(10), 2385–2407.
mla: Browning, Timothy D., et al. “Equidistribution and Freeness on Grassmannians.”
Algebra & Number Theory, vol. 16, no. 10, Mathematical Sciences Publishers,
2022, pp. 2385–407, doi:10.2140/ant.2022.16.2385.
short: T.D. Browning, T. Horesh, F.A. Wilsch, Algebra & 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: '12684'
abstract:
- lang: eng
text: Given a place ω of a global function field K over a finite field, with
associated affine function ring Rω and completion Kω , the aim of this paper
is to give an effective joint equidistribution result for renormalized primitive
lattice points (a,b)∈Rω2 in the plane Kω2 , and for renormalized solutions
to the gcd equation ax+by=1 . The main tools are techniques of Goronik and Nevo
for counting lattice points in well-rounded families of subsets. This gives a
sharper analog in positive characteristic of a result of Nevo and the first author
for the equidistribution of the primitive lattice points in \ZZ2 .
acknowledgement: "The authors warmly thank Amos Nevo for having presented the authors
to each other during\r\na beautiful conference in Goa in February 2016, where the
idea of this paper was born. The\r\nfirst author thanks the IHES for two post-doctoral
years when most of this paper was discussed,\r\nand the Topology team in Orsay for
financial support at the final stage. The first author was\r\nsupported by the EPRSC
EP/P026710/1 grant. Finally, we warmly thank the referee for many\r\nvery helpful
comments that have improved the readability of this paper."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Tal
full_name: Horesh, Tal
id: C8B7BF48-8D81-11E9-BCA9-F536E6697425
last_name: Horesh
- first_name: Frédéric
full_name: Paulin, Frédéric
last_name: Paulin
citation:
ama: Horesh T, Paulin F. Effective equidistribution of lattice points in positive
characteristic. Journal de Theorie des Nombres de Bordeaux. 2022;34(3):679-703.
doi:10.5802/JTNB.1222
apa: Horesh, T., & Paulin, F. (2022). Effective equidistribution of lattice
points in positive characteristic. Journal de Theorie Des Nombres de Bordeaux.
Centre Mersenne. https://doi.org/10.5802/JTNB.1222
chicago: Horesh, Tal, and Frédéric Paulin. “Effective Equidistribution of Lattice
Points in Positive Characteristic.” Journal de Theorie Des Nombres de Bordeaux.
Centre Mersenne, 2022. https://doi.org/10.5802/JTNB.1222.
ieee: T. Horesh and F. Paulin, “Effective equidistribution of lattice points in
positive characteristic,” Journal de Theorie des Nombres de Bordeaux, vol.
34, no. 3. Centre Mersenne, pp. 679–703, 2022.
ista: Horesh T, Paulin F. 2022. Effective equidistribution of lattice points in
positive characteristic. Journal de Theorie des Nombres de Bordeaux. 34(3), 679–703.
mla: Horesh, Tal, and Frédéric Paulin. “Effective Equidistribution of Lattice Points
in Positive Characteristic.” Journal de Theorie Des Nombres de Bordeaux,
vol. 34, no. 3, Centre Mersenne, 2022, pp. 679–703, doi:10.5802/JTNB.1222.
short: T. Horesh, F. Paulin, Journal de Theorie Des Nombres de Bordeaux 34 (2022)
679–703.
date_created: 2023-02-26T23:01:02Z
date_published: 2022-01-27T00:00:00Z
date_updated: 2023-08-04T10:41:40Z
day: '27'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.5802/JTNB.1222
external_id:
arxiv:
- '2001.01534'
isi:
- '000926504300003'
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language:
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license: https://creativecommons.org/licenses/by-nd/4.0/
month: '01'
oa: 1
oa_version: Published Version
page: 679-703
publication: Journal de Theorie des Nombres de Bordeaux
publication_identifier:
eissn:
- 2118-8572
issn:
- 1246-7405
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publisher: Centre Mersenne
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title: Effective equidistribution of lattice points in positive characteristic
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...