Duke Mathematical Journal

Optimal strong approximation for quadratic forms

Naser T. Sardari

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Abstract

For a nondegenerate integral quadratic form F(x1,,xd) in d5 variables, we prove an optimal strong approximation theorem. Let Ω be a fixed compact subset of the affine quadric F(x1,,xd)=1 over the real numbers. Take a small ball B of radius 0<r<1 inside Ω, and an integer m. Further assume that N is a given integer which satisfies Nδ,Ω(r1m)4+δ for any δ>0. Finally assume that an integral vector (λ1,,λd) mod m is given. Then we show that there exists an integral solution x=(x1,,xd) of F(x)=N such that xiλimodm and xNB, provided that all the local conditions are satisfied. We also show that 4 is the best possible exponent. Moreover, for a nondegenerate integral quadratic form in four variables, we prove the same result if N is odd and Nδ,Ω(r1m)6+ϵ. Based on our numerical experiments on the diameter of LPS Ramanujan graphs and the expected square-root cancellation in a particular sum that appears in Remark 6.8, we conjecture that the theorem holds for any quadratic form in four variables with the optimal exponent 4.

Article information

Source
Duke Math. J., Volume 168, Number 10 (2019), 1887-1927.

Dates
Received: 6 April 2017
Revised: 23 January 2019
First available in Project Euclid: 20 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1558339351

Digital Object Identifier
doi:10.1215/00127094-2019-0007

Mathematical Reviews number (MathSciNet)
MR3983294

Subjects
Primary: 11E25: Sums of squares and representations by other particular quadratic forms
Secondary: 11P55: Applications of the Hardy-Littlewood method [See also 11D85]

Keywords
number theory sums of squares quadratic forms applications of the Hardy–Littlewood method

Citation

Sardari, Naser T. Optimal strong approximation for quadratic forms. Duke Math. J. 168 (2019), no. 10, 1887--1927. doi:10.1215/00127094-2019-0007. https://projecteuclid.org/euclid.dmj/1558339351


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