---
_id: '10765'
abstract:
- lang: eng
  text: We establish the Hardy-Littlewood property (à la Borovoi-Rudnick) for Zariski
    open subsets in affine quadrics of the form q(x1,...,xn)=m, where q is a non-degenerate
    integral quadratic form in  n>3 variables and m is a non-zero integer. This gives
    asymptotic formulas for the density of integral points taking coprime polynomial
    values, which is a quantitative version of the arithmetic purity of strong approximation
    property off infinity for affine quadrics.
acknowledgement: "We are grateful to Mikhail Borovoi, Zeev Rudnick and Olivier Wienberg
  for their interest in our\r\nwork. We would like to address our gratitude to Ulrich
  Derenthal for his generous support at Leibniz Universitat Hannover. We are in debt
  to Tim Browning for an enlightening discussion and to the anonymous referees for
  critical comments, which lead to overall improvements of various preliminary versions
  of this paper. Part of this work was carried out and reported during a visit to
  the University of Science and Technology of China. We thank Yongqi Liang for offering
  warm hospitality. The first author was supported by a Humboldt-Forschungsstipendium.
  The second author was supported by grant DE 1646/4-2 of the Deutsche Forschungsgemeinschaft."
article_number: '108236'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Yang
  full_name: Cao, Yang
  last_name: Cao
- first_name: Zhizhong
  full_name: Huang, Zhizhong
  id: 21f1b52f-2fd1-11eb-a347-a4cdb9b18a51
  last_name: Huang
citation:
  ama: Cao Y, Huang Z. Arithmetic purity of the Hardy-Littlewood property and geometric
    sieve for affine quadrics. <i>Advances in Mathematics</i>. 2022;398(3). doi:<a
    href="https://doi.org/10.1016/j.aim.2022.108236">10.1016/j.aim.2022.108236</a>
  apa: Cao, Y., &#38; Huang, Z. (2022). Arithmetic purity of the Hardy-Littlewood
    property and geometric sieve for affine quadrics. <i>Advances in Mathematics</i>.
    Elsevier. <a href="https://doi.org/10.1016/j.aim.2022.108236">https://doi.org/10.1016/j.aim.2022.108236</a>
  chicago: Cao, Yang, and Zhizhong Huang. “Arithmetic Purity of the Hardy-Littlewood
    Property and Geometric Sieve for Affine Quadrics.” <i>Advances in Mathematics</i>.
    Elsevier, 2022. <a href="https://doi.org/10.1016/j.aim.2022.108236">https://doi.org/10.1016/j.aim.2022.108236</a>.
  ieee: Y. Cao and Z. Huang, “Arithmetic purity of the Hardy-Littlewood property and
    geometric sieve for affine quadrics,” <i>Advances in Mathematics</i>, vol. 398,
    no. 3. Elsevier, 2022.
  ista: Cao Y, Huang Z. 2022. Arithmetic purity of the Hardy-Littlewood property and
    geometric sieve for affine quadrics. Advances in Mathematics. 398(3), 108236.
  mla: Cao, Yang, and Zhizhong Huang. “Arithmetic Purity of the Hardy-Littlewood Property
    and Geometric Sieve for Affine Quadrics.” <i>Advances in Mathematics</i>, vol.
    398, no. 3, 108236, Elsevier, 2022, doi:<a href="https://doi.org/10.1016/j.aim.2022.108236">10.1016/j.aim.2022.108236</a>.
  short: Y. Cao, Z. Huang, Advances in Mathematics 398 (2022).
date_created: 2022-02-20T23:01:30Z
date_published: 2022-03-26T00:00:00Z
date_updated: 2023-08-02T14:24:18Z
day: '26'
department:
- _id: TiBr
doi: 10.1016/j.aim.2022.108236
external_id:
  arxiv:
  - '2003.07287'
  isi:
  - '000792517300014'
intvolume: '       398'
isi: 1
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2003.07287
month: '03'
oa: 1
oa_version: Preprint
publication: Advances in Mathematics
publication_identifier:
  eissn:
  - 1090-2082
  issn:
  - 0001-8708
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: Arithmetic purity of the Hardy-Littlewood property and geometric sieve for
  affine quadrics
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 398
year: '2022'
...