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.
