Duke Mathematical Journal

Optimal strong approximation for quadratic forms

Naser T. Sardari

Abstract

For a nondegenerate integral quadratic form $F(x_{1},\dots ,x_{d})$ in $d\geq 5$ variables, we prove an optimal strong approximation theorem. Let $\Omega$ be a fixed compact subset of the affine quadric $F(x_{1},\dots ,x_{d})=1$ over the real numbers. Take a small ball $B$ of radius $0\lt r\lt 1$ inside $\Omega$, and an integer $m$. Further assume that $N$ is a given integer which satisfies $N\gg _{\delta ,\Omega }(r^{-1}m)^{4+\delta }$ for any $\delta \gt 0$. Finally assume that an integral vector $(\lambda _{1},\dots ,\lambda _{d})$ mod $m$ is given. Then we show that there exists an integral solution $\mathbf{x}=(x_{1},\dots ,x_{d})$ of $F(\mathbf{x})=N$ such that $x_{i}\equiv \lambda_{i}\ \mathrm{mod}\ m$ and $\frac{\mathbf{x}}{\sqrt{N}}\in B$, 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 $N\gg _{\delta ,\Omega }(r^{-1}m)^{6+\epsilon }$. 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

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