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arXiv · 2211.13363

Dimensions of Furstenberg sets and an extension of Bourgain's projection theorem

Pablo Shmerkin·Hong Wang
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Abstract

We show that the Hausdorff dimension of $(s,t)$-Furstenberg sets is at least $s+t/2+\epsilon$, where $\epsilon>0$ depends only on $s$ and $t$. This improves the previously best known bound for $2s<t\le 1+\epsilon(s,t)$, in particular providing the first improvement since 1999 to the dimension of classical $s$-Furstenberg sets for $s<1/2$. We deduce this from a corresponding discretized incidence bound under minimal non-concentration assumptions, that simultaneously extends Bourgain's discretized projection and sum-product theorems. The proofs are based on a recent discretized incidence bound of T.~Orponen and the first author and a certain duality between $(s,t)$ and $(t/2,s+t/2)$-Furstenberg sets.

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BibTeXRIS

Pablo Shmerkin, Hong Wang. 2022-11-24. Dimensions of Furstenberg sets and an extension of Bourgain's projection theorem. https://doi.org/10.2140/apde.2025.18.265

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