  Let $\left(a_{n}\right)_{n}$ be a strictly increasing sequence of positive
integers, denote by $A_{N}=\left\{ a_{n}:\,n\leq N\right\} $ its truncations,
and let $\alpha\in\left[0,1\right]$. We prove that if the additive energy
$E\left(A_{N}\right)$ of $A_{N}$ is in $\Omega\left(N^{3}\right)$, then the
sequence $\left(\left\langle \alpha a_{n}\right\rangle \right)_{n}$ of
fractional parts of $\alpha a_{n}$ does not have Poissonian pair correlations
(PPC) for almost every $\alpha$ in the sense of Lebesgue measure. Conversely,
it is known that
$E\left(A_{N}\right)=\mathcal{O}\left(N^{3-\varepsilon}\right)$, for some fixed
$\varepsilon>0$, implies that $\left(\left\langle \alpha a_{n}\right\rangle
\right)_{n}$ has PPC for almost every $\alpha$. This note makes a contribution
to investigating the energy threshold for $E\left(A_{N}\right)$ to imply this
metric distribution property. We establish, in particular, that there exist
sequences $\left(a_{n}\right)_{n}$ with \[
E\left(A_{N}\right)=\Theta\left(\frac{N^{3}}{\log\left(N\right)\log\left(\log
N\right)}\right) \] such that the set of $\alpha$ for which $\left(\alpha
a_{n}\right)_{n}$ does not have PPC is of full Lebesgue measure. Moreover, we
show that for any fixed $\varepsilon>0$ there are sequences
$\left(a_{n}\right)_{n}$ with
$E\left(A_{N}\right)=\Theta\left(\frac{N^{3}}{\log\left(N\right)\left(\log\log
N\right)^{1+\varepsilon}}\right)$ satisfying that the set of $\alpha$ for which
the sequence $\left(\bigl\langle\alpha a_{n}\bigr\rangle\right)_{n}$ does not
have PPC is of full Hausdorff dimension.
