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

We show that the Hausdorff dimension of $(s,t)$-Furstenberg sets is at least $s+t\mathrm{/}2+\epsilon$, 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 $s<1\mathrm{/}2$. 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 $(t\mathrm{/}2,s+t\mathrm{/}2)$-Furstenberg sets.