Carbery, Anthony, Iliopoulou, Marina (2020) Joints formed by lines and a kplane, and a discrete estimate of Kakeya type. Discrete Analysis, 18 . EISSN 23973129. (doi:10.19086/da.18361) (KAR id:85214)
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Official URL https://arxiv.org/abs/1911.09019 
Abstract
Let \(\mathcal{L}\) be a family of lines and let \(\mathcal{P}\) be a family of \(k\)planes in \(\mathbb{F}\)\(^n\) where \(\mathbb{F}\) is a field. In our first result we show that the number of joints formed by a \(k\)plane in \(\mathcal{P}\) together with \({(nk)}\) lines in \(\mathcal{L}\) is \(\mathcal{O}\)\(_n\)(\(\mathcal{L}\)\(\mathcal{P}\)\(^{1/(nk)}\). This is the first sharp result for joints involving higherdimensional affine subspaces, and it holds in the setting of arbitrary fields \(\mathbb{F}\). In contrast, for our second result, we work in the threedimensional Euclidean space \(\mathbb{R}\)\(^3\), and we establish the Kakeyatype estimate
$$\sum_{x \in J} \left(\sum_{\ell \in \mathcal{L}} \chi_\ell(x)\right)^{3/2} \lesssim \mathcal{L}^{3/2}$$
where \(J\) is the set of joints formed by \(\mathcal{L}\); such an estimate fails in the setting of arbitrary fields. This result strengthens the known estimates for joints, including those counting multiplicities. Additionally, our techniques yield significant structural information on quasiextremisers for this inequality.
Item Type:  Article 

DOI/Identification number:  10.19086/da.18361 
Uncontrolled keywords:  Combinatorics; Classical Analysis and ODEs 
Subjects: 
Q Science > QA Mathematics (inc Computing science) > QA165 Combinatorics Q Science > QA Mathematics (inc Computing science) > QA299 Analysis, Calculus 
Divisions:  Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science 
Depositing User:  Marina Iliopoulou 
Date Deposited:  26 Dec 2020 16:58 UTC 
Last Modified:  16 Feb 2021 14:17 UTC 
Resource URI:  https://kar.kent.ac.uk/id/eprint/85214 (The current URI for this page, for reference purposes) 
Iliopoulou, Marina:  https://orcid.org/0000000155379693 
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