On the pair correlation statistic of sequences with the finite gap property

The limiting function f(s) of the pair correlation

$$\begin{aligned} \frac{1}{N} \# \left\{ 1 \le i\ne j\le N \bigg \vert \left\Vert x_i - x_j \right\Vert \le \frac{s}{N} \right\} \end{aligned}$$

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 \(f(s) \ne 2s\), 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.