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

Kornélia Héra; Pablo Shmerkin; Alexia Yavicoli

doi:10.4171/rmi/1281

JournalsrmiVol. 38, No. 1pp. 295–322

An improved bound for the dimension of $(\alpha,2\alpha)$ -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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Abstract

We show that given $\alpha \in (0, 1)$ there is a constant $c=c(\alpha) > 0$ such that any planar $(\alpha, 2\alpha)$ -Furstenberg set has Hausdorff dimension at least $2\alpha + c$ . 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.

Cite this article

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

DOI 10.4171/RMI/1281