The Furstenberg set and its random version
Aihua Fan
Central China Normal University, Wuhan, China; University of Picardie, Amiens, FranceHervé Queffélec
Université de Lille, Villeneuve d’Ascq, FranceMartine Queffélec
Université de Lille, Villeneuve d’Ascq, France
Abstract
We study some number-theoretic, ergodic and harmonic analysis properties of the Furstenberg set of integers and compare them to those of its random analogue . In this half-expository work, we show for example that is “Khinchin distributed”, is far from being Hartman uniformly distributed while is, also that is a -set for all and that is a -Rider set for all such that . Measure-theoretic and probabilistic techniques, notably martingales, play an important role in this work.
Cite this article
Aihua Fan, Hervé Queffélec, Martine Queffélec, The Furstenberg set and its random version. Enseign. Math. 70 (2024), no. 1/2, pp. 61–120
DOI 10.4171/LEM/1040