m -point correlations of the fractional parts of αnθ

Let m≥ 3, we prove that (α n^θ 1)n>0 has Poissonian m-point correlation for all α>0, provided θ<θₘ, where θₘ 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.