Poissonian Pair Correlation for αnθ mod 1

Abstract

We show that sequences of the form $\alpha n^{\theta} \pmod{1}$ with $\alpha > 0$ and $0 < \theta < \tfrac{43}{117} = \tfrac{1}{3} + 0.0341 \ldots$ have Poissonian pair correlation. This improves upon the previous result by Lutsko, Sourmelidis, and Technau, where this was established for $\alpha > 0$ and $0 < \theta < \tfrac{14}{41} = \tfrac{1}{3} + 0.0081 \ldots$.

We reduce the problem of establishing Poissonian pair correlation to a counting problem using a form of amplification and the Bombieri-Iwaniec double large sieve. The counting problem is then resolved non-optimally by appealing to the bounds of Robert-Sargos and (Fouvry-Iwaniec-)Cao-Zhai. The exponent $\theta = \tfrac{2}{5}$ is the limit of our approach.