# On the number of gaps of sequences with Poissonian Pair Correlations

## TL;DR

This paper investigates the structure of sequences with Poissonian pair correlations, showing that the maximum multiplicity of neighboring gap lengths grows slower than linearly, and constructing sequences with controlled gap diversity.

## Contribution

It improves understanding of gap multiplicities in sequences with Poissonian pair correlations and constructs sequences with bounded gap diversity, answering a previously open question.

## Key findings

- Maximum multiplicity of neighboring gaps is o(n)
- Sequences with Poissonian pair correlations can have bounded gap multiplicities
- Answers negatively a question by G. Larcher about gap diversity

## Abstract

A sequence $(x_n)$ on the torus is said to have Poissonian pair correlations if $\# \{1\le i\neq j\le N: |x_i-x_j| \le s/N\}=2sN(1+o(1))$ 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 $\{x_1,\ldots,x_n\}$ 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 $\{x_1,\ldots,x_n\}$ 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.