Arithmetic purity of the Hardy-Littlewood property and geometric sieve for affine quadrics

Cao, Yang; Huang, Zhizhong

Arithmetic purity of the Hardy-Littlewood property and geometric sieve for affine quadrics

Cao Y, Huang Z. 2022. Arithmetic purity of the Hardy-Littlewood property and geometric sieve for affine quadrics. Advances in Mathematics. 398(3), 108236.

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https://arxiv.org/abs/2003.07287 [Preprint]

DOI

10.1016/j.aim.2022.108236

Journal Article | Published | English

Scopus indexed

Author

Cao, Yang; Huang, Zhizhong^ISTA

Department

Browning Group

Abstract

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.

Publishing Year

2022

Date Published

2022-03-26

Journal Title

Advances in Mathematics

Publisher

Elsevier

Acknowledgement

We are grateful to Mikhail Borovoi, Zeev Rudnick and Olivier Wienberg for their interest in our work. 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.

Volume

398

Issue

Article Number

108236

ISSN

0001-8708

eISSN

1090-2082

IST-REx-ID

10765

Cite this

Cao Y, Huang Z. Arithmetic purity of the Hardy-Littlewood property and geometric sieve for affine quadrics. Advances in Mathematics. 2022;398(3). doi:10.1016/j.aim.2022.108236

Cao, Y., & Huang, Z. (2022). Arithmetic purity of the Hardy-Littlewood property and geometric sieve for affine quadrics. Advances in Mathematics. Elsevier. https://doi.org/10.1016/j.aim.2022.108236

Cao, Yang, and Zhizhong Huang. “Arithmetic Purity of the Hardy-Littlewood Property and Geometric Sieve for Affine Quadrics.” Advances in Mathematics. Elsevier, 2022. https://doi.org/10.1016/j.aim.2022.108236.

Y. Cao and Z. Huang, “Arithmetic purity of the Hardy-Littlewood property and geometric sieve for affine quadrics,” Advances in Mathematics, vol. 398, no. 3. Elsevier, 2022.

Cao Y, Huang Z. 2022. Arithmetic purity of the Hardy-Littlewood property and geometric sieve for affine quadrics. Advances in Mathematics. 398(3), 108236.

Cao, Yang, and Zhizhong Huang. “Arithmetic Purity of the Hardy-Littlewood Property and Geometric Sieve for Affine Quadrics.” Advances in Mathematics, vol. 398, no. 3, 108236, Elsevier, 2022, doi:10.1016/j.aim.2022.108236.

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