Integrability of orthogonal projections, and applications to Furstenberg sets
Dąbrowski, Damian; Orponen, Tuomas; Villa, Michele (2022-07-15)
Dąbrowski, D., Orponen, T., & Villa, M. (2022). Integrability of orthogonal projections, and applications to Furstenberg sets. Advances in Mathematics, 407, 108567. https://doi.org/10.1016/j.aim.2022.108567
© 2022 The Author(s). This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
https://creativecommons.org/licenses/by/4.0/
https://urn.fi/URN:NBN:fi-fe2022112967578
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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 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 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
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