Integrability of orthogonal projections, and applications to Furstenberg sets 

Author:  Dąbrowski, Damian^{1}; Orponen, Tuomas^{1}; Villa, Michele^{1,2} 
Organizations: 
^{1}Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FI40014 University of Jyväskylä, Finland ^{2}Research Unit of Mathematical Sciences, University of Oulu, P.O. Box 8000, FI90014, University of Oulu, Finland 
Format:  article 
Version:  published version 
Access:  open 
Online Access:  PDF Full Text (PDF, 0.7 MB) 
Persistent link:  http://urn.fi/urn:nbn:fife2022112967578 
Language:  English 
Published: 
Elsevier,
2022

Publish Date:  20221129 
Description: 
AbstractLet \(\mathcal G(d,n)\) 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 \[\int\limits_{\mathcal G(d,n)}\left\left\pi_V\mu\right\right_{L^p(V)}^pd\gamma_{d,n}(V)<\infty, 1\leqslant p<\frac{2dns}{ds}.\] The upper bound for \(p\) is sharp, at least, for \(d1\leqslant s\leqslant d\), 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 \[\dim_{\mathrm H}K\geqslant2s+(1s)(t1).\] 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 codimension1 Furstenberg sets in \(\mathbb{R}^d\). As another corollary of our method, we obtain a \(\delta\)discretised sumproduct estimate for \((\delta,s)\)sets. Our bound improves on a previous estimate of Chen for every \(\frac12<s<1\), and also of GuthKatzZahl for \(s\geqslant0.5151\). see all

Series: 
Advances in mathematics 
ISSN:  00018708 
ISSNE:  10902082 
ISSNL:  00018708 
Volume:  407 
Article number:  108567 
DOI:  10.1016/j.aim.2022.108567 
OADOI:  https://oadoi.org/10.1016/j.aim.2022.108567 
Type of Publication: 
A1 Journal article – refereed 
Field of Science: 
111 Mathematics 
Subjects:  
Copyright information: 
© 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/ 