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This paper is a complement to (Fan et al. in Enseign Math 70:61–120, 2024 [21]). It surveys the works on the Furstenberg set \(S=\{2^{m}3^{n}: m\ge 0, n\ge 0\}\) and its random version T. We also present some new results. For example, it is proved that T almost surely contains a subset of positive lower density which is \(\frac{4}{3}\) -Rider. It is also proved that a class of random sets of integers are Sidon sets as soon as Bourgain’s condition is not satisfied; this generalizes a result of Kahane-Katznelson. Some open questions about S and T are listed at the end of the paper.

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Old and New Results on the Furstenberg Sets

  • 会议论文

  • 发表日期：

    2025年

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  • Aihua Fan,
  • Hervé Queffélec,
  • Martine Queffélec

摘要

This paper is a complement to (Fan et al. in Enseign Math 70:61–120, 2024 [21]). It surveys the works on the Furstenberg set \(S=\{2^{m}3^{n}: m\ge 0, n\ge 0\}\) and its random version T. We also present some new results. For example, it is proved that T almost surely contains a subset of positive lower density which is \(\frac{4}{3}\) -Rider. It is also proved that a class of random sets of integers are Sidon sets as soon as Bourgain’s condition is not satisfied; this generalizes a result of Kahane-Katznelson. Some open questions about S and T are listed at the end of the paper.