Optimal strong approximation for quadratic forms

Abstract

For a non-degenerate integral quadratic form F (x 1 , . . . , x d ) in d ≥ 5 variables, we prove an optimal strong approximation theorem. Let Ω be a fixed compact subset of the affine quadric F (x 1 , . . . , x d ) = 1 over the real numbers. Take 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 ≫ δ,Ω (r −1 m) 4+δ for any δ > 0. Finally assume that an integral vector (λ 1 , . . . , λ d ) mod m is given. Then we show that there exists an integral solution X = (x 1 , . . . , x d ) of F (X) = N such that x i ≡ λ i mod m and X √ 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 in 4 variables we prove the same result if N is odd and N ≫ δ,Ω (r −1 m) 6+ǫ . 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 4 variables with the optimal exponent 4.

Diameter of Ramanujan Graphs and Random Cayley Graphs

“…The numerical experiments suggest that the diameter of the (number theoretic) LPS Ramanujan Graphs is asymptotic to (1.2) (4/3) log 5 (n). This is consistent with our conjecture on the optimal strong approximation for quadratic forms in 4 variables [Sar15a]. On the other hand, the numerical data suggests that the diameter of the random Cayley graph equals that of a random 6-regular graph [BFdlV82], that is (1.3) log 5 (n).…”

Twisted Linnik implies Optimal Covering Exponent for $S^3$

“…One of the key innovations in Sardari's work [6] concerns the introduction of a new basis given by the tangent space of F at ξ and we proceed to recall the construction here. Let e 4 = ξ.…”

“…Unfortunately, the unconditional estimates obtained in [10] are not sharp enough to prove that that K(S 3 ) < 2 unconditionally. Using Sardari's work [6] as a base, we shall establish the following result. Theorem 1.2.…”

Quantitative results on Diophantine equations in many variables

“…However, for generic quadratic forms one can do much better [BD08]. Recently, Sardari proved an optimal strong approximation theorem for f − N, where f is a non-degenerate quadratic form and N a sufficiently large integer [Sar15].…”