Pair correlation densities of inhomogeneous quadratic forms II

TL;DR

This paper proves that the local pair correlation density of certain inhomogeneous quadratic forms follows Poisson statistics under specific diophantine conditions, supporting a conjecture about spectral correlations in integrable systems.

Contribution

It establishes Poissonian pair correlation densities for inhomogeneous quadratic forms in multiple dimensions under diophantine conditions, extending previous results and confirming a conjecture in spectral theory.

Findings

Poisson statistics hold for all diophantine vectors in 2D.

In higher dimensions, Poisson behavior occurs only for vectors of certain diophantine types.

Supports the Berry-Tabor conjecture on spectral correlations in integrable systems.

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 $\kappa<(k-1)/(k-2)$. Our findings support a conjecture of Berry and Tabor on the Poisson nature of spectral correlations in quantized integrable systems.