On the number of gaps of sequences with Poissonian Pair Correlations

A sequence $(x_n)$ on the torus is said to have Poissonian pair correlations if $\mathrm{\#}\{1\le i\mathrm{\ne }j\le N:\mathrm{\mid }{x}_i-x_j\mathrm{\mid }\le s\mathrm{/}N\}=2sN(1+o(1))$ for all reals $s>0$, as $N\to \mathrm{\infty }$. It is known that, if $(x_n)$ has Poissonian pair correlations, then the number $g(n)$ of different gap lengths between neighboring elements of $\{x_1,\dots ,x_n\}$ cannot be bounded along every index subsequence $(n_t)$. First, we improve this by showing that the maximum among the multiplicities of the neighboring gap lengths of $\{x_1,\dots ,x_n\}$ is $o(n)$, as $n\to \infty$\mathrm{.}Furthermore,weshowthat,foreveryfunction$f: $\mathbf{N}$^+\to $\mathbf{N}$^+$with$\lim_n f(n)=\infty$,thereexistsasequence$(x_n)$ with Poissonian pair correlations and such that $g(n)\le f(n)$ for all sufficiently large $n$. This answers negatively a question posed by G. Larcher.