Additive energy and the metric poissonian property

Aled Walker (University of Oxford)

Frank Adams 1,

Let A be a set of natural numbers. The metric poissonian property, which concerns the distribution of dilates of A modulo 1, was first introduced to pure mathematics by Rudnick-Sarnak in 1998, motivated by applications to quantum mechanics. It has recently received renewed attention, owing to the discovery of a strong link between the additive energy of A (a crude measure of the additive structure of A) and whether or not A enjoys the metric poissonian property. In this talk we will discuss our work on the quantitative dependence between these two notions, and will formulate a precise conjecture on the true relationship, namely that there exists a sharp Khintchine-type threshold: that is, that the metric Poissonian property should be completely determined by whether or not a certain sum of additive energies is convergent or divergent.
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