Pair correlation densities of inhomogeneous quadratic forms II

2002/10/14 by Jens Marklof, Marklof, Jens · 1 citation
Mathematics · Physics and Astronomy · #Analytic Number Theory Research #Chaotic Dynamics (nlin.CD) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Approximation and Integration #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Number Theory (math.NT) #math-ph #math.DS #math.MP #math.NT #nlin.CD

paper · pdf · doi:10.48550/arxiv.math/0210198

arxiv created 2002/10/14 · openalex publication_date 2002/10/14 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Denote by ‖ ⋅ ‖ the euclidean norm in \RR^k. We prove that the local pair correlation density of the sequence ‖ \vecm -\vecalf ‖^k, \vecm∈\ZZ^k, is that of a Poisson process, under diophantine conditions on the fixed vector \vecalf∈\RR^k: in dimension two, vectors \vecalf of any diophantine type are admissible; in higher dimensions (k>2), Poisson statistics are only observed for diophantine vectors of type κ

Citations

• Level clustering in the regular spectrum

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• A survey and a result on inhomogeneous quadratic forms

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