Abstract
We show that sequences of the form αnθ (mod 1) with α > 0 and 0 < θ < 11743 = 13 + 0.0341 . . . have Poissonian pair correlation. This improves upon the previous result by Lutsko, Sourmelidis, and Technau, where this was established for α > 0 and 0 < θ < 1441 = 13 + 0.0081 . . .. 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 θ = 25 is the limit of our approach.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 7654-7679 |
| Number of pages | 26 |
| Journal | International Mathematics Research Notices |
| Volume | 2024 |
| Issue number | 9 |
| DOIs | |
| State | Published - May 1 2024 |
Funding
This work was supported by the National Science Foundation grant [DMS-1902063 to M.R.]; and a joint grant by Fonds zur F\u00F6rderung der wissenschaftlichen Forschung and Agence Nationale de la Recherche [FWF: I 4945-N and ANR-20-CE91-0006 to A.S.]. The second author thanks Christoph Aistleitner, Tim Browning, and members of their research groups for the helpful conversations during his short visits. We also wish to thank the referees for their careful reading of the paper and their comments.
ASJC Scopus subject areas
- General Mathematics