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

### 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
Format: article
Version: published version
Access: open
Online Access: PDF Full Text (PDF, 0.7 MB)
Language: English
Published: Elsevier, 2022
Publish Date: 2022-11-29
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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 $$0<n<d$$.

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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