Full Poissonian Local Statistics of Slowly Growing Sequences

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