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 + (1)(2)t + epsilon, where epsilon > 0 depends only on s and t. This improves the previously best known bound for 2s < t <= 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 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 ((1)(2)t, s + (1)(2)t)-Furstenberg sets.