On the number of gaps of sequences with Poissonian Pair Correlations

Abstract

A sequence $(x_n)$ on the torus is said to have Poissonian pair correlations if $\#\{1\le i\ne j\le N:|x_i-x_j|\le s/N\}=2sN(1+o(1))$ 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 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 $n\to \infty$. 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 and such that $g(n)\le f(n)$ for all sufficiently large $n$. This answers negatively a question posed by G. Larcher.

13 pages, 1 figure, comments are welcome