---
_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. <i>Journal de Theorie des Nombres de Bordeaux</i>. 2022;34(3):679-703.
    doi:<a href="https://doi.org/10.5802/JTNB.1222">10.5802/JTNB.1222</a>
  apa: Horesh, T., &#38; Paulin, F. (2022). Effective equidistribution of lattice
    points in positive characteristic. <i>Journal de Theorie Des Nombres de Bordeaux</i>.
    Centre Mersenne. <a href="https://doi.org/10.5802/JTNB.1222">https://doi.org/10.5802/JTNB.1222</a>
  chicago: Horesh, Tal, and Frédéric Paulin. “Effective Equidistribution of Lattice
    Points in Positive Characteristic.” <i>Journal de Theorie Des Nombres de Bordeaux</i>.
    Centre Mersenne, 2022. <a href="https://doi.org/10.5802/JTNB.1222">https://doi.org/10.5802/JTNB.1222</a>.
  ieee: T. Horesh and F. Paulin, “Effective equidistribution of lattice points in
    positive characteristic,” <i>Journal de Theorie des Nombres de Bordeaux</i>, 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.” <i>Journal de Theorie Des Nombres de Bordeaux</i>,
    vol. 34, no. 3, Centre Mersenne, 2022, pp. 679–703, doi:<a href="https://doi.org/10.5802/JTNB.1222">10.5802/JTNB.1222</a>.
  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'
file:
- access_level: open_access
  checksum: 08f28fded270251f568f610cf5166d69
  content_type: application/pdf
  creator: dernst
  date_created: 2023-02-27T09:10:13Z
  date_updated: 2023-02-27T09:10:13Z
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  file_size: 870468
  relation: main_file
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file_date_updated: 2023-02-27T09:10:13Z
has_accepted_license: '1'
intvolume: '        34'
isi: 1
issue: '3'
language:
- iso: eng
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
publication_status: published
publisher: Centre Mersenne
quality_controlled: '1'
scopus_import: '1'
status: public
title: Effective equidistribution of lattice points in positive characteristic
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  name: Creative Commons Attribution-NoDerivatives 4.0 International (CC BY-ND 4.0)
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type: journal_article
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volume: 34
year: '2022'
...
---
_id: '2904'
abstract:
- lang: eng
  text: Generalized van der Corput sequences are onedimensional, infinite sequences
    in the unit interval. They are generated from permutations in integer base b and
    are the building blocks of the multi-dimensional Halton sequences. Motivated by
    recent progress of Atanassov on the uniform distribution behavior of Halton sequences,
    we study, among others, permutations of the form P(i) = ai (mod b) for coprime
    integers a and b. We show that multipliers a that either divide b - 1 or b + 1
    generate van der Corput sequences with weak distribution properties. We give explicit
    lower bounds for the asymptotic distribution behavior of these sequences and relate
    them to sequences generated from the identity permutation in smaller bases, which
    are, due to Faure, the weakest distributed generalized van der Corput sequences.
- lang: fre
  text: Les suites de Van der Corput généralisées sont dessuites unidimensionnelles
    et infinies dans l’intervalle de l’unité.Elles sont générées par permutations
    des entiers de la basebetsont les éléments constitutifs des suites multi-dimensionnelles
    deHalton. Suites aux progrès récents d’Atanassov concernant le com-portement de
    distribution uniforme des suites de Halton nous nousintéressons aux permutations
    de la formuleP(i)  =ai(modb)pour les entiers premiers entre euxaetb. Dans cet
    article nousidentifions des multiplicateursagénérant des suites de Van derCorput
    ayant une mauvaise distribution. Nous donnons les bornesinférieures explicites
    pour cette distribution asymptotique asso-ciée à ces suites et relions ces dernières
    aux suites générées parpermutation d’identité, qui sont, selon Faure, les moins
    bien dis-tribuées des suites généralisées de Van der Corput dans une basedonnée.
article_processing_charge: No
article_type: original
author:
- first_name: Florian
  full_name: Pausinger, Florian
  id: 2A77D7A2-F248-11E8-B48F-1D18A9856A87
  last_name: Pausinger
  orcid: 0000-0002-8379-3768
citation:
  ama: Pausinger F. Weak multipliers for generalized van der Corput sequences. <i>Journal
    de Theorie des Nombres des Bordeaux</i>. 2012;24(3):729-749. doi:<a href="https://doi.org/10.5802/jtnb.819">10.5802/jtnb.819</a>
  apa: Pausinger, F. (2012). Weak multipliers for generalized van der Corput sequences.
    <i>Journal de Theorie Des Nombres Des Bordeaux</i>. Université de Bordeaux. <a
    href="https://doi.org/10.5802/jtnb.819">https://doi.org/10.5802/jtnb.819</a>
  chicago: Pausinger, Florian. “Weak Multipliers for Generalized van Der Corput Sequences.”
    <i>Journal de Theorie Des Nombres Des Bordeaux</i>. Université de Bordeaux, 2012.
    <a href="https://doi.org/10.5802/jtnb.819">https://doi.org/10.5802/jtnb.819</a>.
  ieee: F. Pausinger, “Weak multipliers for generalized van der Corput sequences,”
    <i>Journal de Theorie des Nombres des Bordeaux</i>, vol. 24, no. 3. Université
    de Bordeaux, pp. 729–749, 2012.
  ista: Pausinger F. 2012. Weak multipliers for generalized van der Corput sequences.
    Journal de Theorie des Nombres des Bordeaux. 24(3), 729–749.
  mla: Pausinger, Florian. “Weak Multipliers for Generalized van Der Corput Sequences.”
    <i>Journal de Theorie Des Nombres Des Bordeaux</i>, vol. 24, no. 3, Université
    de Bordeaux, 2012, pp. 729–49, doi:<a href="https://doi.org/10.5802/jtnb.819">10.5802/jtnb.819</a>.
  short: F. Pausinger, Journal de Theorie Des Nombres Des Bordeaux 24 (2012) 729–749.
date_created: 2018-12-11T12:00:15Z
date_published: 2012-01-01T00:00:00Z
date_updated: 2023-10-18T07:53:47Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.5802/jtnb.819
file:
- access_level: open_access
  checksum: 6954bfe9d7f4119fbdda7a11cf0f5c67
  content_type: application/pdf
  creator: dernst
  date_created: 2020-05-11T12:40:39Z
  date_updated: 2020-07-14T12:45:52Z
  file_id: '7819'
  file_name: JTNB_2012__24_3_729_0.pdf
  file_size: 819275
  relation: main_file
file_date_updated: 2020-07-14T12:45:52Z
has_accepted_license: '1'
intvolume: '        24'
issue: '3'
language:
- iso: eng
month: '01'
oa: 1
oa_version: Published Version
page: 729 - 749
publication: Journal de Theorie des Nombres des Bordeaux
publication_identifier:
  eissn:
  - 2118-8572
  issn:
  - 1246-7405
publication_status: published
publisher: Université de Bordeaux
publist_id: '3843'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Weak multipliers for generalized van der Corput sequences
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 24
year: '2012'
...