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.

We investigate the statistical behavior of the eigenvalues and diameter of random Cayley graphs of SL 2 [Z/pZ] as the prime number p goes to infinity. We prove a density theorem for the number of exceptional eigenvalues of random Cayley graphs i.e. the eigenvalues with absolute value bigger than the optimal spectral bound. Our numerical results suggest that random Cayley graphs of SL 2 [Z/pZ] and the explicit LPS Ramanujan projective graphs of P 1 (Z/pZ) have optimal spectral gap and diameter as the prime number p goes to infinity.

We study the diameter of LPS Ramanujan graphs Xp,q. We show that the diameter of the bipartite Ramanujan graphs is greater than (4/3) log p (n) + O(1) where n is the number of vertices of Xp,q. We also construct an infinite family of (p + 1)-regular LPS Ramanujan graphs Xp,m such that the diameter of these graphs is greater than or equal to (4/3) log p (n) . On the other hand, for any k-regular Ramanujan graph we show that the distance of only a tiny fraction of all pairs of vertices is greater than (1 + ) log k−1 (n). We also have some numerical experiments for LPS Ramanujan graphs and random Cayley graphs which suggest that the diameters are asymptotically (4/3) log k−1 (n) and log k−1 (n), respectively.

By assuming some widely believed arithmetic conjectures, we show that the task of accepting a number that is representable as a sum of $d\geq 2$ squares subjected to given congruence conditions is NP-complete. On the other hand, we develop and implement a deterministic polynomial-time algorithm that represents a number as a sum of four squares with some restricted congruence conditions, by assuming a polynomial-time algorithm for factoring integers and Conjecture 1.1. As an application, we develop and implement a deterministic polynomial-time algorithm for navigating Lubotzky, Phillips, Sarnak (LPS) Ramanujan graphs, under the same assumptions.

Let Q(X) be any integral primitive positive definite quadratic form in k variables where k ≥ 4 and discriminant D. We give an upper bound on the number of integral solutions of Q(X) = n for any integer n in terms of n, k and D. As a corollary, we give a definite answer to a conjecture of Lester and Rudnick on the small scale equidistribution of orthonormal basis of eigenfunctions restricted to an individual eigenspace on the flat torus T d for d ≥ 5. Another application of our main theorem gives a sharp upper bound on A d (n, t), the number of representation of the positive definite quadratic form Q(x, y) = nx 2 + 2txy + ny 2 as a sum of squares of d ≥ 5 binary linear forms where n − n 1 (d−1) −o(1) < t < n. This upper bound allows us to study the local statistics of integral points on sphere.