Kornélia Héra, Pablo Shmerkin, Alexia Yavicoli
Published 2020-01-30Version 1
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.
On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane
New estimates on the size of $(α,2α)$-Furstenberg sets
On the distance sets spanned by sets of dimension $d/2$ in $\mathbb{R}^d$
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