  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.
