Additive energy and the metric Poissonian property

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 a1+a2=a3+a4 with ai∈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 Khinchin-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.