Optimal strong approximation for quadratic forms

TL;DR

This paper proves an optimal strong approximation theorem for integral quadratic forms in five or more variables, establishing conditions under which integral solutions approximate points on the associated affine quadric with prescribed congruences.

Contribution

It introduces the first optimal strong approximation results for quadratic forms in five or more variables and extends similar results to four variables under specific conditions.

Findings

Proves optimal strong approximation for $d \\geq 5$ variables.

Establishes the best possible exponent 4 for the approximation.

Provides partial results and conjectures for the four-variable case.

Abstract

For a non-degenerate integral quadratic form $F(x_1, \dots , x_d)$ in $d\geq5$ 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 $0<r<1$ 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 $\delta>0$. Finally assume that an integral vector $(\lambda_1, \dots, \lambda_d) $ mod $m$ is given. Then we show that there exists an integral solution $X=(x_1,\dots,x_d)$ of $F(X)=N$ such that $x_i\equiv \lambda_i \text{ mod } m$ and $\frac{X}{\sqrt{N}}\in B$, provided that all the local conditions are satisfied. We also show that 4 is the best possible exponent. Moreover, for a non-degenerate integral quadratic form in 4 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~\ref{evidence}, we conjecture that the theorem holds for any quadratic form in 4 variables with the optimal exponent $4$.