---
_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. <i>Pacific Journal of Mathematics</i>.
    2023;324(2):265-294. doi:<a href="https://doi.org/10.2140/pjm.2023.324.265">10.2140/pjm.2023.324.265</a>'
  apa: 'Horesh, T., &#38; Nevo, A. (2023). Horospherical coordinates of lattice points
    in hyperbolic spaces: Effective counting and equidistribution. <i>Pacific Journal
    of Mathematics</i>. Mathematical Sciences Publishers. <a href="https://doi.org/10.2140/pjm.2023.324.265">https://doi.org/10.2140/pjm.2023.324.265</a>'
  chicago: 'Horesh, Tal, and Amos Nevo. “Horospherical Coordinates of Lattice Points
    in Hyperbolic Spaces: Effective Counting and Equidistribution.” <i>Pacific Journal
    of Mathematics</i>. Mathematical Sciences Publishers, 2023. <a href="https://doi.org/10.2140/pjm.2023.324.265">https://doi.org/10.2140/pjm.2023.324.265</a>.'
  ieee: 'T. Horesh and A. Nevo, “Horospherical coordinates of lattice points in hyperbolic
    spaces: Effective counting and equidistribution,” <i>Pacific Journal of Mathematics</i>,
    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.” <i>Pacific Journal
    of Mathematics</i>, vol. 324, no. 2, Mathematical Sciences Publishers, 2023, pp.
    265–94, doi:<a href="https://doi.org/10.2140/pjm.2023.324.265">10.2140/pjm.2023.324.265</a>.'
  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:
- access_level: open_access
  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
  success: 1
file_date_updated: 2023-09-05T07:26:17Z
has_accepted_license: '1'
intvolume: '       324'
isi: 1
issue: '2'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
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)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 324
year: '2023'
...
---
_id: '12793'
abstract:
- lang: eng
  text: "Let F be a global function field with constant field Fq. Let G be a reductive
    group over Fq. We establish a variant of Arthur's truncated kernel for G and for
    its Lie algebra which generalizes Arthur's original construction. We establish
    a coarse geometric expansion for our variant truncation.\r\nAs applications, we
    consider some existence and uniqueness problems of some cuspidal automorphic representations
    for the functions field of the projective line P1Fq with two points of ramifications."
acknowledgement: 'I’d like to thank Prof. Chaudouard for introducing me to this area.
  I’d like to thank Prof. Harris for asking me the question that makes Section 10
  possible. I’m grateful for the support of Prof. Hausel and IST Austria. The author
  was funded by an ISTplus fellowship: This project has received funding from the
  European Union’s Horizon 2020 research and innovation programme under the Marie
  Skłodowska-Curie Grant Agreement No. 754411.'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Hongjie
  full_name: Yu, Hongjie
  id: 3D7DD9BE-F248-11E8-B48F-1D18A9856A87
  last_name: Yu
  orcid: 0000-0001-5128-7126
citation:
  ama: Yu H.  A coarse geometric expansion of a variant of Arthur’s truncated traces
    and some applications. <i>Pacific Journal of Mathematics</i>. 2022;321(1):193-237.
    doi:<a href="https://doi.org/10.2140/pjm.2022.321.193">10.2140/pjm.2022.321.193</a>
  apa: Yu, H. (2022).  A coarse geometric expansion of a variant of Arthur’s truncated
    traces and some applications. <i>Pacific Journal of Mathematics</i>. Mathematical
    Sciences Publishers. <a href="https://doi.org/10.2140/pjm.2022.321.193">https://doi.org/10.2140/pjm.2022.321.193</a>
  chicago: Yu, Hongjie. “ A Coarse Geometric Expansion of a Variant of Arthur’s Truncated
    Traces and Some Applications.” <i>Pacific Journal of Mathematics</i>. Mathematical
    Sciences Publishers, 2022. <a href="https://doi.org/10.2140/pjm.2022.321.193">https://doi.org/10.2140/pjm.2022.321.193</a>.
  ieee: H. Yu, “ A coarse geometric expansion of a variant of Arthur’s truncated traces
    and some applications,” <i>Pacific Journal of Mathematics</i>, vol. 321, no. 1.
    Mathematical Sciences Publishers, pp. 193–237, 2022.
  ista: Yu H. 2022.  A coarse geometric expansion of a variant of Arthur’s truncated
    traces and some applications. Pacific Journal of Mathematics. 321(1), 193–237.
  mla: Yu, Hongjie. “ A Coarse Geometric Expansion of a Variant of Arthur’s Truncated
    Traces and Some Applications.” <i>Pacific Journal of Mathematics</i>, vol. 321,
    no. 1, Mathematical Sciences Publishers, 2022, pp. 193–237, doi:<a href="https://doi.org/10.2140/pjm.2022.321.193">10.2140/pjm.2022.321.193</a>.
  short: H. Yu, Pacific Journal of Mathematics 321 (2022) 193–237.
date_created: 2023-04-02T22:01:11Z
date_published: 2022-08-29T00:00:00Z
date_updated: 2023-08-04T10:42:38Z
day: '29'
department:
- _id: TaHa
doi: 10.2140/pjm.2022.321.193
ec_funded: 1
external_id:
  arxiv:
  - '2109.10245'
  isi:
  - '000954466300006'
intvolume: '       321'
isi: 1
issue: '1'
keyword:
- Arthur–Selberg trace formula
- cuspidal automorphic representations
- global function fields
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2109.10245
month: '08'
oa: 1
oa_version: Preprint
page: 193-237
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '754411'
  name: ISTplus - Postdoctoral Fellowships
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: ' A coarse geometric expansion of a variant of Arthur''s truncated traces and
  some applications'
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 321
year: '2022'
...