On the number of gaps and Poissonian pair correlations

A sequence (xn) on the unit interval is said to have Poissonian pair correlation if #{1≤i≠j≤N:‖xi−xj‖≤s/N}=2sN(1+o(1)) for all reals s>0, as N→∞. It is known that, if (xn) has Poissonian pair correlations, then the number g(n) of different gap lengths between neighboring elements of {x1,…,xn} cannot be bounded along any index subsequence (nt). First, we improve this by showing that, if (xn) has Poissonian pair correlations, then the maximum among the multiplicities of the neighboring gap lengths of {x1,…,xn} is o(n), as n→∞. Furthermore, we show that for every function f:N+→N+ with limn⁡f(n)=∞ there exists a sequence (xn) with Poissonian pair correlations such that g(n)≤f(n) for all sufficiently large n. This answers negatively a question posed by G. Larcher.