Integrability of orthogonal projections, and applications to Furstenberg sets

Abstract

Let $\mathcal{G}(d,n)$ be the Grassmannian manifold of $n$-dimensional subspaces of ${\mathbb{R}}^d$, and let ${\pi }_V:{\mathbb{R}}^d\to V$ be the orthogonal projection. We prove that if $\mu$ is a compactly supported Radon measure on ${\mathbb{R}}^d$ satisfying the $s$-dimensional Frostman condition $\mu (B(x,r))\le C{r}^s$ for all $x\in {\mathbb{R}}^d$ and $r>0$, then $${\int }_{\mathcal{G}(d,n)}|{\pi }_V\mu {|}_{L^p(V)}^p,d{\gamma }_{d,n}(V)<\mathrm{\infty },1\le p<\frac{2d-n-s}{d-s}.$$ The upper bound for $p$ is sharp, at least, for $d-1\le s\le d$, 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 $0\le s\le 1$ and $0\le t\le 2$, a set $K\subset {\mathbb{R}}^2$ is called an $(s,t)$-Furstenberg set if there exists a $t$-dimensional family $\mathcal{L}$ of affine lines in ${\mathbb{R}}^2$ such that ${\dim }_{\mathrm{H}}(K\cap \ell )\ge s$ for all $\ell \in \mathcal{L}$. As a consequence of our projection theorem in ${\mathbb{R}}^2$, we show that every $(s,t)$-Furstenberg set $K\subset {\mathbb{R}}^2$ with $1<t\le 2$ satisfies $${\dim }_{\mathrm{H}}K\ge 2s+(1-s)(t-1).$$ This improves on previous bounds for pairs $(s,t)$ with $s>\frac{1}{2}$ and $t\ge 1+ϵ$ for a small absolute constant $ϵ>0$. We also prove an analogue of this estimate for $(d-1,s,t)$-Furstenberg sets in ${\mathbb{R}}^d$.