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 (Formula presented.), where ϵ > 0 depends only on s and t. This improves the previously best known bound for 2s < t ≤ 1 + ϵ(s, t), in particular providing the first improvement since 1999 to the dimension of classical s-Furstenberg sets for (Formula presented.). We deduce this from a corresponding discretized incidence bound under minimal nonconcentration 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 (Formula presented.)-Furstenberg sets.