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The n-point correlation of quadratic forms

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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 \(\left( v_{1},\ldots ,v_{n}\right) \), with bounded norm, such that the \(n-1\) differences \(Q\left( v_{1}\right) -Q\left( v_{2}\right) ,\ldots Q\left( v_{n-1}\right) -Q\left( v_{n}\right) \), 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 (Distribution of values of quadratic forms at integral points. http://www.math.uni-bielefeld.de/sfb701/files/preprints/sfb13003.pdf, 2013) of Götze and 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.

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  1. Mathematics Department Technion, Israel Institute of Technology, 32000, Haifa, Israel

    Oliver Sargent

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  1. Oliver Sargent
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Correspondence to Oliver Sargent.

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Communicated by S. G. Dani.

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Sargent, O. The n-point correlation of quadratic forms. Monatsh Math 178, 259–297 (2015). https://doi.org/10.1007/s00605-015-0817-5

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  • DOI: https://doi.org/10.1007/s00605-015-0817-5

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