Joint IAS/Princeton University Number Theory Seminar

For a non-degenerate integral quadratic form $F(x_1, \dots , x_d)$ in 5 (or more) variables, we prove an optimal strong approximation theorem. Fix any compact subspace $\Omega\subset\mathbb{R}^d$ of the affine quadric $F(x_1,\dots,x_d)=1$. 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 $N\gg(r^{-1}m)^{4+\epsilon}$ for any $\epsilon > 0$. Finally assume that we are given an integral vector $(\lambda_1, \dots, \lambda_d) $ mod $m$. 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 $F(x_1, \dots , x_4)$ in 4 variables we prove the same result if $N\geq (r^{-1}m)^{6+\epsilon}$ and some non-singular local conditions for $N$ are satisfied. 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 $4$.