University of Oulu

Dąbrowski, D., Orponen, T., & Villa, M. (2022). Integrability of orthogonal projections, and applications to Furstenberg sets. Advances in Mathematics, 407, 108567.

Integrability of orthogonal projections, and applications to Furstenberg sets

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Author: Dąbrowski, Damian1; Orponen, Tuomas1; Villa, Michele1,2
Organizations: 1Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland
2Research Unit of Mathematical Sciences, University of Oulu, P.O. Box 8000, FI-90014, University of Oulu, Finland
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Version: published version
Access: open
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Language: English
Published: Elsevier, 2022
Publish Date: 2022-11-29


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 \(s\geqslant0.5151\).

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Series: Advances in mathematics
ISSN: 0001-8708
ISSN-E: 1090-2082
ISSN-L: 0001-8708
Volume: 407
Article number: 108567
DOI: 10.1016/j.aim.2022.108567
Type of Publication: A1 Journal article – refereed
Field of Science: 111 Mathematics
Copyright information: © 2022 The Author(s). This is an open access article under the CC BY license (