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For a non-degenerate integral quadratic form F(x1,...,xd) in 5 (or more) variables, we prove an optimal strong approximation theorem. Fix any compact subspace Ω⊂Rd of the affine quadric F(x1,...,xd)=1. Suppose that we are given a small ball B of radius 0 < r < 1 inside Ω, and an integer m. Further assume that N is a given integer which satisfies N ≫ (r−1m)4+ε for any ε > 0. Finally assume that we are given an integral vector (λ1, . . . , λd) mod m. Then we show that there exists an integral solution x = (x1, . . . , xd) x of F(x)=N such that xi ≡λi mod m and √N ∈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 (x1 , . . . , x4 ) in 4 variables we prove the same result if N ≥ (r−1m)6+ε and N is not divisible by 2k where 2k ≫ Nε for any ε. 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.

• CategoryNumber Theory
• Duration: 01:49:46
• Date:  October 14, 2015 at 1:25 PM
• Views: 109
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