An improved bound for the dimension of (α,2α)(\alpha,2\alpha)-Furstenberg sets

  • Kornélia Héra

    The University of Chicago, USA
  • Pablo Shmerkin

    Universidad Torcuato Di Tella, Ciudad de Buenos Aires, Argentina
  • Alexia Yavicoli

    The University of British Columbia, Vancouver, Canada
An improved bound for the dimension of $(\alpha,2\alpha)$-Furstenberg sets cover
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We show that given α(0,1)\alpha \in (0, 1) there is a constant c=c(α)>0c=c(\alpha) > 0 such that any planar (α,2α)(\alpha, 2\alpha)-Furstenberg set has Hausdorff dimension at least 2α+c2\alpha + c. This improves several previous bounds, in particular extending a result of Katz–Tao and Bourgain. We follow the Katz–Tao approach with suitable changes, along the way clarifying, simplifying and/or quantifying many of the steps.

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Kornélia Héra, Pablo Shmerkin, Alexia Yavicoli, An improved bound for the dimension of (α,2α)(\alpha,2\alpha)-Furstenberg sets. Rev. Mat. Iberoam. 38 (2022), no. 1, pp. 295–322

DOI 10.4171/RMI/1281