An Explicit non-Poissonian Pair Correlation Function

A generic uniformly distributed random sequence on the unit interval has Poissonian pair correlations. Usually, the pair correlations statistic is therefore studied for equidistributed sequences. At the same time, there are only very few explicitly known examples of sequences with this property and many types of deterministic sequences have been proven to fail having the Poissonian pair correlation property. In this paper we study the pair correlation statistic in the non-uniform case and analyze the first elementary example of such a sequence, namely $x_n := \left\{ \frac{\log(2n-1)}{\log(2)} \right\}$, which is a standard low-dispersion sequence. The proof heavily relies on a full understanding of the gap structure of $(x_n)_{n=1}^N$. Furthermore, we discuss differences to the weak pair correlation function which turns out to be linear.