On the Pair Correlation Statistic of Sequences with Finite Gap Property

The limiting function $f(s)$ of the pair correlation $$\frac{1}{N}\mathrm{\#}\{1\le i\mathrm{\ne }j\le N|\parallel x_i-x_j\parallel \le \frac{s}{N}\}$$ for a sequence $(x_N{)}_{N\in \mathbb{N}}$ on the torus ${\mathbb{T}}^1$ is said to be Poissonian if it exists and equals $2s$ for all $s\ge 0$. For instance, independent, uniformly distributed random variables generically have this property. Obviously $f(s)$ is always a monotonic function if existent. There are only few examples of sequences where $f(s)\mathrm{\ne }2s$, but where the limit can still be explicitly calculated. Therefore, it is an open question which types of functions $f(s)$ can or cannot appear here. In this note, we give a partial answer on this question by addressing the case that the number of different gap lengths in the sequence is finite and showing that $f$ cannot be continuous then.