ADDITIVE ENERGY AND THE METRIC POISSONIAN PROPERTY

Abstract

Let be a set of natural numbers. Recent work has suggested a strong link between the additive energy of (the number of solutions to with ) and the metric Poissonian property, which is a fine-scale equidistribution property for dilates of 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.

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• Mathematika - View - PDF