Optimal strong approximation for quadratic forms

Abstract

For a nondegenerate integral quadratic form $F(x_1,\dots ,x_d)$ in $d\ge 5$ 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 $0<r<1$ 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 $\mathbf{x}=(x_1,\dots ,x_d)$ 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 $N{\gg }_{\delta ,\Omega }(r^{-1}m{)}^{6+ϵ}$. 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 $4$.