Additive energy and the metric Poissonian property

Thomas F Bloom, Sam Chow, Ayla Gafni, Aled Walker

OAI: oai:www.repository.cam.ac.uk:1810/291729 • DOI: 10.17863/CAM.38889

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.

math.NT math.NT math.CO 11J71 11J83 05B10 11B30 60F10