Poissonian Pair Correlation for $\alpha n^{\theta }$ mod 1

Abstract

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.