A sequence $(x_n)$ on the unit interval is said to have Poissonian pair correlation if $\mathrm{\#}\{1\le i\ne j\le N:\Vert x_i-x_j\Vert \le s/N\}=2sN(1+o(1))$ for all reals $s>0$, as $N\to \infty$. 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 any index subsequence $(n_t)$. First, we improve this by showing that, if $(x_n)$ has Poissonian pair correlations, then the maximum among the multiplicities of the neighboring gap lengths of $\{x_1,\dots ,x_n\}$ is $o(n)$, as $n\to \infty$. Furthermore, we show that for every function $f:{\mathbf{N}}^{+}\to {\mathbf{N}}^{+}$ with ${\lim }_{n}f(n)=\infty$ there exists a sequence $(x_n)$ with Poissonian pair correlations such that $g(n)\le f(n)$ for all sufficiently large n. This answers negatively a question posed by G. Larcher.

一个序列 $(X_n)$ 在单位区间上被称为具有泊松对相关性，如果 $\mathrm{\#}\{1\le \mathrm{一世}\ne j\le N：\Vert X_{\mathrm{一世}}-X_j\Vert \le 秒/N\}=2秒N(1+○(1))$ 对于所有实数 $秒>0$， 作为 $N\to \infty$. 据了解，如果$(X_n)$ 具有泊松对相关性，那么数 $G(n)$ 相邻元素之间的不同间隙长度 $\{X_1,\dots ,X_n\}$ 不能沿着任何索引子序列有界 $(n_{吨})$. 首先，我们通过证明，如果$(X_n)$ 具有泊松对相关性，那么相邻间隙长度的多重性中的最大值 $\{X_1,\dots ,X_n\}$ 是 $○(n)$， 作为 $n\to \infty$. 此外，我们证明对于每个函数$F：{\mathbf{N}}^{+}\to {\mathbf{N}}^{+}$ 和 ${\mathrm{林}}_{n}F(n)=\infty$ 存在一个序列 $(X_n)$ 具有泊松对相关性，使得 $G(n)\le F(n)$对于所有足够大的n。这否定了 G. Larcher 提出的问题。