Full Poissonian local statistics of slowly growing sequences

Creators: Lutsko, Christopher¹; Technau, Niclas^{2, 3}

1. University of Houston
2. Max Planck Institute for Mathematics
3. California Institute of Technology

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Abstract

Fix α > 0. Then by Fejér's theorem (α(logn)^A mod1)_(n≥1) is uniformly distributed if and only if A > 1. We sharpen this by showing that all correlation functions, and hence the gap distribution, are Poissonian provided A > 1. This is the first example of a deterministic sequence modulo 1 whose gap distribution and all of whose correlations are proven to be Poissonian. The range of A is optimal and complements a result of Marklof and Strömbergsson who found the limiting gap distribution of (log(n) mod1), which is necessarily not Poissonian.

Copyright and License

This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. Compositio Mathematica is © Foundation Compositio Mathematica.

Acknowledgement

We thank Apoorva Khare, Jens Marklof and Zeev Rudnick for comments on a previous version of the paper. Furthermore, we are grateful to the anonymous referee for a careful reading and comments that helped remove inaccuracies from an earlier version of the paper.

Funding

NT was supported by a Schrödinger Fellowship of the Austrian Science Fund (FWF): project J 4464-N.

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