Correlations of the Fractional Parts of $\alpha n^{\theta }$

Abstract

Let $m\ge 3$, we prove that $(\alpha n^{\theta }\mathrm{mod}1{)}_{n>0}$ has Poissonian $m$-point correlation for all $\alpha >0$, provided $\theta <{\theta }_m$, where ${\theta }_m$ is an explicit bound which goes to $0$ as $m$ increases. This work builds on the method developed in Lutsko-Sourmelidis-Technau (2021), and introduces a new combinatorial argument for higher correlation levels, and new Fourier analytic techniques. A key point is to introduce an `extra' frequency variable to de-correlate the sequence variables and to eventually exploit a repulsion principle for oscillatory integrals. Presently, this is the only positive result showing that the $m$-point correlation is Poissonian for such sequences.

17 Pages