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15 July 2019 Optimal strong approximation for quadratic forms
Naser T. Sardari
Duke Math. J. 168(10): 1887-1927 (15 July 2019). DOI: 10.1215/00127094-2019-0007


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.


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Naser T. Sardari. "Optimal strong approximation for quadratic forms." Duke Math. J. 168 (10) 1887 - 1927, 15 July 2019.


Received: 6 April 2017; Revised: 23 January 2019; Published: 15 July 2019
First available in Project Euclid: 20 May 2019

zbMATH: 07108022
MathSciNet: MR3983294
Digital Object Identifier: 10.1215/00127094-2019-0007

Primary: 11E25
Secondary: 11P55

Keywords: applications of the Hardy–Littlewood method , number theory , Quadratic forms , sums of squares

Rights: Copyright © 2019 Duke University Press


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Vol.168 • No. 10 • 15 July 2019
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