Spread-out Furstenberg sets I: the sticky case

Abstract

We introduce spread-out measures, a notion of local non-concentration near pencils for measures on Grassmannians. We establish an optimal $δ$-discretized dual Furstenberg estimate for ``sticky spread-out'' shadings. A simple-to-state corollary is the following. Assume that $ν$ is an Ahlfors $s$-regular measure on $\mathbb{G}(2,1)$ with $s > 0$. Let $E \subset \mathbb{R}^{2}$ be a Borel set with equal Hausdorff and packing dimension. Then, $\dim_{\mathrm{H}} Π_{V}(E) = \min\{\dim_{\mathrm{H}} E,1\} \quad \text{for $ν$-a.e. } V \in S^1$. The general $δ$-discretized estimate for ``sticky spread out'' configurations here is the starting point for a more general theorem in the companion paper of Cohen, Shmerkin, Wang and Zakharov, where the ``stickiness'' hypothesis is removed. For instance, the assumption of equal Hausdorff and packing dimension in the result stated above will be dispensed with in the companion paper.