  Let $A$ be a set of natural numbers. Recent work has suggested a strong link
between the additive energy of $A$ (the number of solutions to $a_1 + a_2 = a_3
+ a_4$ with $a_i \in A$) and the metric Poissonian property, which is a
fine-scale equidistribution property for dilates of $A$ modulo $1$. There
appears to be reasonable evidence to speculate a sharp Khintchine-type
threshold, that is, to speculate that the metric Poissonian property should be
completely determined by whether or not a certain sum of additive energies is
convergent or divergent. In this article, we primarily address the convergence
theory, in other words the extent to which having a low additive energy forces
a set to be metric Poissonian.
