The Furstenberg set and its random version

  • Aihua Fan

    Central China Normal University, Wuhan, China; University of Picardie, Amiens, France
  • Hervé Queffélec

    Université de Lille, Villeneuve d’Ascq, France
  • Martine Queffélec

    Université de Lille, Villeneuve d’Ascq, France
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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