Spread-out Furstenberg sets II: general configurations and fractal restricted projections

Abstract

We prove optimal dual Furstenberg estimates for Frostman and Katz--Tao configurations for which the shadings are not locally concentrated near pencils. The starting point is the sticky estimate of Orponen and Shmerkin in a companion paper. We deduce restricted projection theorems in the plane and in higher dimensions. In particular, Ahlfors-regular measures on the circle satisfy a Marstrand-type projection theorem. Applications include restricted projections to surfaces in the $\mathbb{G}(4,2)$ Grassmannian, pinned distances, discretized sum-product, and a new proof of the dimension formula for planar self-affine sets.