Optimal strong approximation for quadratic forms

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 $4$.