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15 August 2020 The density of rational points near hypersurfaces
Jing-Jing Huang
Duke Math. J. 169(11): 2045-2077 (15 August 2020). DOI: 10.1215/00127094-2020-0004

Abstract

We establish a sharp asymptotic formula for the number of rational points up to a given height and within a given distance from a hypersurface. Our main innovation is a bootstrap method that relies on the synthesis of Poisson summation, projective duality, and the method of stationary phase. This has surprising applications to counting rational points lying on the manifold; indeed, we are able to prove an analogue of Serre’s dimension growth conjecture (originally stated for projective varieties) in this general setup. As another consequence of our main counting result, we solve the generalized Baker–Schmidt problem in the simultaneous setting for hypersurfaces.

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Jing-Jing Huang. "The density of rational points near hypersurfaces." Duke Math. J. 169 (11) 2045 - 2077, 15 August 2020. https://doi.org/10.1215/00127094-2020-0004

Information

Received: 22 March 2018; Revised: 4 January 2020; Published: 15 August 2020
First available in Project Euclid: 3 July 2020

MathSciNet: MR4132580
Digital Object Identifier: 10.1215/00127094-2020-0004

Subjects:
Primary: 11J83
Secondary: 11J13, 11J25

Rights: Copyright © 2020 Duke University Press

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Vol.169 • No. 11 • 15 August 2020
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