1 December 2002 Pair correlation densities of inhomogeneous quadratic forms, II
Jens Marklof
Duke Math. J. 115(3): 409-434 (1 December 2002). DOI: 10.1215/S0012-7094-02-11531-2


Denote by $\parallel\cdot\parallel$ the Euclidean norm in $\mathbb {R}\sp k$. We prove that the local pair correlation density of the sequence $\parallel\mathbf {m}-\mathbf {\alpha}\parallel\sp k,\mathbf {m}\in \mathbb {Z}\sp k$, is that of a Poisson process, under Diophantine conditions on the fixed vector $\mathbf {\alpha}\in \mathbb {R}\sp k$ in dimension two, vectors $\mathbf {\alpha}$ of any Diophantine type are admissible; in higher dimensions $(k>2)$, Poisson statistics are observed only for Diophantine vectors of type $\kappa<(k-1)/(k-2)$. Our findings support a conjecture of M. Berry and M. Tabor on the Poisson nature of spectral correlations in quantized integrable systems.


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Jens Marklof. "Pair correlation densities of inhomogeneous quadratic forms, II." Duke Math. J. 115 (3) 409 - 434, 1 December 2002. https://doi.org/10.1215/S0012-7094-02-11531-2


Published: 1 December 2002
First available in Project Euclid: 26 May 2004

zbMATH: 1136.11325
MathSciNet: MR1940408
Digital Object Identifier: 10.1215/S0012-7094-02-11531-2

Primary: 11P21
Secondary: 11Fxx , 37Jxx , 37N20 , 81Q10

Rights: Copyright © 2002 Duke University Press

Vol.115 • No. 3 • 1 December 2002
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