Poissonian pair correlation for $\alpha n^{\theta }$

Abstract

We show that sequences of the form $\alpha n^{\theta }(\mathrm{mod}1)$ with $\alpha >0$ and $0<\theta <\frac{43}{117}=\frac{1}{3}+0.0341\dots$ have Poissonian pair correlation. This improves upon the previous result by Lutsko, Sourmelidis, and Technau, where this was established for $\alpha >0$ and $0<\theta <\frac{14}{41}=\frac{1}{3}+0.0081\dots$. 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 $\theta =\frac{2}{5}$ is the limit of our approach.

35 pages, 3 figures