Full Poissonian local statistics of slowly growing sequences

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