Optimal strong approximation for quadratic forms

For a non-degenerate integral quadratic form $F(x_1,…,x_d)$ in 5
(or more) variables, we prove an optimal strong approximation theorem. Fix
any compact subspace $\Omega\subset R^d$ of the affine quadric $F(x_1,…,x_d)=1$. Suppose
that we are given 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 $N (r−1m)4+\epsilon$
for any $\epsilon 0$. Finally assume that we are given an integral vector $(\lambda_1, … , \lambda_d)
mod m$. Then we show that there exists an integral solution $x = (x_1, … , x_d)$
of $F(x)=N$ such that $x_i =\lambda_i mod m$ and $\sqrt{N} \in B$,provided that all the local conditions are satisfied. We also show that 4 is the best possible expo-
nent. Moreover, for a non-degenerate integral quadratic form $F (x_1 , … , x_4 )$ in 4 variables we prove the same result if $N \ge (r-1)6+\epsilon$ and N is not divisible by $ 2k$ where $2k N\epsilon$ for any $\epsilon$. Based on some numerical experiments on the diameter of LPS Ramanujan graphs, we conjecture that the optimal strong approximation theorem holds for any quadratic form $F(X) $ in 4 variables with the optimal exponent 4.