Integrability of orthogonal projections, and applications to Furstenberg sets

Abstract: Let $\mathcal{G}(d,n)$ be the Grassmannian manifold of $n$-dimensional subspaces of $\mathbb{R}<sup>{d}$, and let $\pi_{V} \colon \mathbb{R}^{d} \to V$ be the orthogonal projection. We prove that if $\mu$ is a compactly supported Radon measure on $\mathbb{R}<sup>{d}$ satisfying the $s$-dimensional Frostman condition $\mu(B(x,r)) \leq Cr<sup>{s}$ for all $x \in \mathbb{R}<sup>{d}$ and $r &gt; 0$, then $$\int_{\mathcal{G}(d,n)} |\pi_{V}\mu|<em>{L^{{p}(V)}^{p}} \, d\gamma</em>{d,n}(V) &lt; \infty, \qquad 1 \leq p &lt; \frac{2d - n - s}{d - s}.$$ The upper bound for $p$ is sharp, at least, for $d - 1 \leq s \leq d$, and every $0 < n < d$. Our motivation for this question comes from finding improved lower bounds on the Hausdorff dimension of $(s,t)$-Furstenberg sets. For $0 \leq s \leq 1$ and $0 \leq t \leq 2$, a set $K \subset \mathbb{R}<sup>{2}$ is called an $(s,t)$-Furstenberg set if there exists a $t$-dimensional family $\mathcal{L}$ of affine lines in $\mathbb{R}<sup>{2}$ such that $\dim_{\mathrm{H}} (K \cap \ell) \geq s$ for all $\ell \in \mathcal{L}$. As a consequence of our projection theorem in $\mathbb{R}<sup>{2}$, we show that every $(s,t)$-Furstenberg set $K \subset \mathbb{R}<sup>{2}$ with $1 &lt; t \leq 2$ satisfies $\dim_{\mathrm{H}} K \geq 2s + (1 - s)(t - 1).$ This improves on previous bounds for pairs $(s,t)$ with $s &gt; \tfrac{1}{2}$ and $t \geq 1 + \epsilon$ for a small absolute constant $\epsilon &gt; 0$. We also prove a higher dimensional analogue of this estimate for codimension-1 Furstenberg sets in $\mathbb{R}<sup>{d}$. As another corollary of our method, we obtain a $\delta$-discretised sum-product estimate for $(\delta,s)$-sets. Our bound improves on a previous estimate of Chen for every $\tfrac{1}{2} &lt; s &lt; 1$, and also of Guth-Katz-Zahl for $s \geq 0.5151$.