For a nondegenerate integral quadratic form F.x1;:::;xd/ in d 5 variables, we prove an optimal strong approximation theorem. Let be a fixed compact subset of the affine quadric F.x1;:::;xd/ 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/4Cı 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;:::;xd/ of F.x/ D N such that xi 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.
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