Pair correlation for quadratic polynomials mod 1

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.

Pair correlation and equidistribution on manifolds

“…For instance (1.2) is known to hold for ξ j = 〈 j k α〉 (k ≥ 2 a fixed integer) for Lebesgue-almost every α [16], and a lower bound on the Haussdorff dimension of permissible α has recently been established [3]. But so far there is not a single explicit example of α, such as α = 2 or α = π, for which (1.2) holds; not even in the quadratic case k = 2 [9,13,14]. There has been significant recent progess in characterising the Poisson pair correlation (1.2) for general sequences ξ j = 〈a j α〉, for Lebesgue-almost every α, in terms of the additive energy of the integer coefficients a j ; cf.…”

Global eigenvalue distribution of matrices defined by the skew-shift

“…• Rudnick, Sarnak, and Zaharescu [25] conjectured that skew-shift orbits exhibit Poissonian spacing (as i.i.d. sequences would), and in fact proved this along subsequences for topologically generic frequencies; see also [13,20,21,24]. By contrast, the spacing distribution of irrational circle rotation displays level repulsion [5,23].…”

A pair correlation problem, and counting lattice points with the zeta function

“…A notable exception is the sequence ( √ n) n∈Z ≥1 \ , which is known to have Poissonian pair correlation [16]. The sequence (n 2 α) n≥1 is conjectured to have Poissonian pair correlation under mild Diophantine assumptions on α, but only partial results are known in this direction [20,29,33,41]. Lacking specific examples, it is natural to turn to a metric theory instead.…”