Poissonian pair correlation for specific polynomial rotations

Establish Poissonian pair correlation for a specific individual irrational value of \(\alpha\), such as \(\alpha=\sqrt{2}\), for polynomial sequences \((n^d\alpha)_{n\geq1}\) with degree \(d\geq2\).

Background

The paper contrasts metric results, which establish Poissonian pair correlation for almost every parameter α\alpha, with the unresolved problem of proving the property for any prescribed individual irrational parameter. The example α=2\alpha=\sqrt{2} is given explicitly. This issue concerns pair correlation rather than the random number-variance problem solved in the paper.

References

However, a significant gap remains: while these and related results give a relatively complete picture of Poissonian behavior in the metrical sense (i.e., for a set of \alpha of full Lebesgue measure), the problem remains wide open for any specific individual value of \alpha such as \alpha = \sqrt{2}, see .

— On the asymptotic behavior of the number variance of random points on the unit torus  (2610.11687 - Aistleitner et al., 8 Oct 2026) in Section 1, Introduction and statement of results