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

Kornélia Héra; Pablo Shmerkin; Alexia Yavicoli

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

Classical Analysis and ODEs 2024-08-19 v2 Combinatorics Metric Geometry

Authors: Kornélia Héra , Pablo Shmerkin , Alexia Yavicoli

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.

Keywords

furstenberg sets hausdorff dimension katz-tao method planar sets geometric measure theory

Cite

@article{arxiv.2001.11304,
  title  = {An improved bound for the dimension of $(\alpha,2\alpha)$-Furstenberg sets},
  author = {Kornélia Héra and Pablo Shmerkin and Alexia Yavicoli},
  journal= {arXiv preprint arXiv:2001.11304},
  year   = {2024}
}

Comments

29 pages. v2: many small corrections, results unchanged

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

Abstract

Keywords

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