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 $a_{1}+a_{2}=a_{3}+a_{4}$ 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 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.