Duke Mathematical Journal
- Duke Math. J.
- Volume 168, Number 10 (2019), 1887-1927.
Optimal strong approximation for quadratic forms
For a nondegenerate integral quadratic form in variables, we prove an optimal strong approximation theorem. Let be a fixed compact subset of the affine quadric over the real numbers. Take a small ball of radius inside , and an integer . Further assume that is a given integer which satisfies for any . Finally assume that an integral vector mod is given. Then we show that there exists an integral solution of such that and , provided that all the local conditions are satisfied. We also show that is the best possible exponent. Moreover, for a nondegenerate integral quadratic form in four variables, we prove the same result if is odd and . 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 .
Duke Math. J., Volume 168, Number 10 (2019), 1887-1927.
Received: 6 April 2017
Revised: 23 January 2019
First available in Project Euclid: 20 May 2019
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Mathematical Reviews number (MathSciNet)
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