Full Poissonian Local Statistics of Slowly Growing Sequences

Abstract

Fix $\alpha >0$, then by Fejér's theorem $(\alpha (\log n{)}^A\mathrm{mod}1{)}_{n\ge 1}$ 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 one 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 $(\log (n)\mathrm{mod}1)$, which is necessarily not Poissonian.

Minor edits, 23 pages, 1 figure

References in corpus (1)

• On the sign patterns of entrywise positivity preservers in fixed dimension