Optimal strong approximation for quadratic forms

For a nondegenerate integral quadratic form F.x₁;:::;x_d/ in d 5 variables, we prove an optimal strong approximation theorem. Let be a fixed compact subset of the affine quadric F.x₁;:::;x_d/ D 1 over the real numbers. Take a small ball B of radius 0 < r < 1 inside , and an integer m. Further assume that N is a given integer which satisfies N _ı; .r¹m/⁴C^ı for any ı > 0. Finally assume that an integral vector .₁;:::;_d/ mod m is given. Then we show that there exists an integral solution x D .x₁;:::;x_d/ of F.x/ D N such that x_{i i} mod m and ^px _N 2 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 _ı; .r¹m/⁶C. 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.