Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $\# \{1\le i\neq j\le N: |x_i-x_j| \le s/N\}=2sN(1+o(1))$ 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.

18
The categories of this paper are: econ
29
The categories of this paper are: funct-an
31
The categories of this paper are: ao-sci
12
The categories of this paper are: nucl-th
11
The categories of this paper are: hep-lat
1
The categories of this paper are: cs
9
The categories of this paper are: nlin
27
The categories of this paper are: comp-gas
13
The categories of this paper are: quant-ph
25
The categories of this paper are: patt-sol
17
The categories of this paper are: q-fin
14
The categories of this paper are: math-ph
19
The categories of this paper are: eess
24
The categories of this paper are: dg-ga
10
The categories of this paper are: q-bio
32
The categories of this paper are: atom-ph