# Pair correlation for quadratic polynomials mod 1

## TL;DR

This paper establishes explicit Diophantine conditions under which the pair correlation statistics of quadratic polynomial fractional parts at integers converge to a Poisson distribution, supporting the Berry-Tabor conjecture in quantum chaos.

## Contribution

It provides the first explicit Diophantine conditions for quadratic polynomials ensuring Poissonian pair correlation statistics.

## Key findings

- Convergence of pair correlation density under specified conditions
- Support for the Berry-Tabor conjecture in quantum chaos
- Explicit criteria for quadratic polynomial coefficients

## Abstract

It is an open question whether the fractional parts of nonlinear polynomials at integers have the same fine-scale statistics as a Poisson point process. Most results towards an affirmative answer have so far been restricted to almost sure convergence in the space of polynomials of a given degree. We will here provide explicit Diophantine conditions on the coefficients of polynomials of degree 2, under which the convergence of an averaged pair correlation density can be established. The limit is consistent with the Poisson distribution. Since quadratic polynomials at integers represent the energy levels of a class of integrable quantum systems, our findings provide further evidence for the Berry-Tabor conjecture in the theory of quantum chaos.