  We show for sequences $\left(a_{n}\right)_{n \in \mathbb{N}}$ of distinct
positive integers with maximal order of additive energy, that the sequence
$\left(\left\{a_{n} \alpha\right\}\right)_{n \in \mathbb{N}}$ does not have
Poissonian pair correlations for any $\alpha$. This result essentially sharpens
a result obtained by J. Bourgain on this topic.
