Abstract

Let \(\mathcal G(d,n)\) be the Grassmannian manifold of \(n\)-dimensional subspaces of \(\mathbb{R}^d\), and let \(\pi_V : \mathbb{R}^d \rightarrow V\)be the orthogonal projection. We prove that if \(μ\) is a compactly supported Radon measure on \(\mathbb{R}^d\) satisfying the \(s\)-dimensional Frostman condition \(\mu(B(x,r))\leqslant Cr^s\) for all \(x\in\mathbb{R}^d\) and \(r>0\), then \[\int\limits_{\mathcal G(d,n)}\left|\left|\pi_V\mu\right|\right|_{L^p(V)}^pd\gamma_{d,n}(V)<\infty, 1\leqslant p<\frac{2d-n-s}{d-s}.\]

The upper bound for \(p\) is sharp, at least, for \(d-1\leqslant s\leqslant d\), 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 \(0\leqslant s\leqslant1\) and \(0\leqslant t\leqslant2\), 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 \(\dim_{\mathrm H}(K\cap\ell)\geqslant s\) 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 \(1<t\leqslant2\) satisfies \[\dim_{\mathrm H}K\geqslant2s+(1-s)(t-1).\]

This improves on previous bounds for pairs \((s,t)\) with \(s>\frac12\) and \(t\geqslant1+\epsilon\) for a small absolute constant \(\epsilon>0\). We also prove a higher dimensional analogue of this estimate for codimension-1 Furstenberg sets in \(\mathbb{R}^d\). 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 \(\frac12<s<1\), and also of Guth-Katz-Zahl for \(s\geqslant0.5151\).