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

TL;DR

This paper improves lower bounds on the Hausdorff dimension of Furstenberg sets, extending Bourgain's projection theorem by establishing new discretized incidence bounds under minimal assumptions.

Contribution

It introduces a novel lower bound for the dimension of Furstenberg sets and extends Bourgain's discretized projection and sum-product theorems using recent incidence bounds.

Findings

Hausdorff dimension of $(s,t)$-Furstenberg sets is at least $s+t/2+ ext{epsilon}$

First improvement since 1999 for classical $s$-Furstenberg sets with $s<1/2$

Extended Bourgain's discretized projection and sum-product theorems

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.