Poissonian correlations of $\alpha n^d$ mod $1$

TL;DR

This paper proves that the fractional parts of $\alpha n^d$ exhibit Poissonian correlations for most $\alpha$ when $d$ is large, aligning with the Berry--Tabor conjecture, and also for certain badly approximable $\alpha$.

Contribution

It establishes Poissonian $\ell$-point correlations for large $d$ and for a full Hausdorff dimension set of badly approximable $\alpha$, using a novel Fourier analytic and counting approach.

Findings

Poissonian correlations hold for almost all $\alpha$ when $d$ is large.

Poissonian correlations are shown for a full Hausdorff dimension set of badly approximable $\alpha$.

The proof employs a determinant method to count points on diagonal hypersurfaces.

Abstract

Let $x(n):=\alpha n^d \mod 1$ for integer $d >1$ and non-zero real $\alpha$. We show that $\{x(n)\}_{n>0}$ has Poissonian $\ell$-point correlations for almost all choices of $\alpha$ when $d$ is large (depending on $\ell$). This falls in line with the expected behavior from the Berry--Tabor conjecture. Further, in the spirit of a conjecture of Rudnick--Sarnak, we show Poissonian $\ell$-point correlations for a set of badly approximable $\alpha$ of full Hausdorff dimension by a Fourier analytic transference principle. The proof makes use of an application of the determinant method to count points on a diagonal hypersurface of degree $d$ in such a way as to capture the contribution of points belonging to lower dimensional varieties. As $d$ grows, these `special solutions' dominate the count and non-special solutions become increasingly rare. This stratified counting statement allows us to control the number of points on average very effectively.