  We say that a sequence $(x_n)_{n \in \mathbb{N}}$ in $[0,1)$ has Poissonian
pair correlations if \begin{equation*} \lim_{N \to \infty} \frac{1}{N} \# \left
\lbrace 1 \leq l \neq m \leq N: \| x_l - x_m \| \leq \frac{s}{N} \right \rbrace
= 2s \end{equation*} for every $s \geq 0$. The aim of this article is twofold.
First, we will establish a gap theorem which allows to deduce that a sequence
$(x_n)_{n \in \mathbb{N}}$ of real numbers in $[0,1)$ having a certain weak gap
structure, cannot have Poissonian pair correlations. This result covers a broad
class of sequences, e.g., Kronecker sequences, the van der Corput sequence and
in more general $LS$-sequences of points and digital $(t,1)$-sequences.
Additionally, this theorem enables us to derive negative pair correlation
properties for sequences of the form $(\lbrace a_n \alpha \rbrace)_{n \in
\mathbb{N}}$, where $(a_n)_{n \in \mathbb{N}}$ is a strictly increasing
sequence of integers with maximal order of additive energy, a notion that plays
an important role in many fields, e.g., additive combinatorics, and is strongly
connected to Poissonian pair correlation problems. These statements are not
only metrical results, but hold for all possible choices of $\alpha$.
  Second, in this note we study the pair correlation statistics for sequences
of the form, $x_n = \lbrace b^n \alpha \rbrace, \ n=1, 2, 3, \ldots$, with an
integer $b \geq 2$, where we choose $\alpha$ as the Stoneham number and as an
infinite de Bruijn word. We will prove that both instances fail to have the
Poissonian property. Throughout this article $\lbrace \cdot \rbrace$ denotes
the fractional part of a real number.
