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Pair correlation densities of inhomogeneous quadratic forms II

Jens Marklof
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Abstract

Denote by $\| \cdot \|$ the euclidean norm in $\RR^k$. We prove that the local pair correlation density of the sequence $\| \vecm -\vecalf \|^k$, $\vecm\in\ZZ^k$, is that of a Poisson process, under diophantine conditions on the fixed vector $\vecalf\in\RR^k$: 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 $κ<(k-1)/(k-2)$. Our findings support a conjecture of Berry and Tabor on the Poisson nature of spectral correlations in quantized integrable systems.

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Jens Marklof. 2002-10-14. Pair correlation densities of inhomogeneous quadratic forms II. https://arxiv.org/abs/math/0210198

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