The n-point correlation of quadratic forms

Abstract

In this paper we investigate the distribution of the set of values of a quadratic form Q, at integral points. In particular we are interested in the n-point correlations of the this set. The asymptotic behaviour of the counting function that counts the number of n-tuples of integral points $\left(v_1,\dots ,v_n\right)$, with bounded norm, such that the n-1 differences $Q\left(v_1\right)-Q\left(v_2\right),\dots Q\left(v_{n-1}\right)-Q\left(v_n\right)$, lie in prescribed intervals is obtained. The results are valid provided that the quadratic form has rank at least 5, is not a multiple of a rational form and n is at most the rank of the quadratic form. For certain quadratic forms satisfying Diophantine conditions we obtain a rate for the limit. The proofs are based on those in the recent preprint ([G-M]) of F. G$\ddot{o}$tze and G. Margulis, in which they prove an `effective' version of the Oppenheim Conjecture. In particular, the proofs rely on Fourier analysis and estimates for certain theta series.