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.


Journal Article | Epub ahead of print | English

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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
Date Published
2022-02-21
Journal Title
Advances in Mathematics
Acknowledgement
We are grateful to Mikhail Borovoi, Zeev Rudnick and Olivier Wi‹enberg 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
3
Article Number
108236
ISSN
eISSN
IST-REx-ID

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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