Some negative results related to Poissonian pair correlation problems

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.