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Joints formed by lines and a k-plane, and a discrete estimate of Kakeya type

Carbery, Anthony, Iliopoulou, Marina (2020) Joints formed by lines and a k-plane, and a discrete estimate of Kakeya type. Discrete Analysis, 18 . E-ISSN 2397-3129. (doi:10.19086/da.18361) (KAR id:85214)

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 \({(n-k)}\) lines in \(\mathcal{L}\) is \(\mathcal{O}\)\(_n\)(|\(\mathcal{L}\)||\(\mathcal{P}\)|\(^{1/(n-k)}\). This is the first sharp result for joints involving higher-dimensional affine subspaces, and it holds in the setting of arbitrary fields \(\mathbb{F}\). In contrast, for our second result, we work in the three-dimensional Euclidean space \(\mathbb{R}\)\(^3\), and we establish the Kakeya-type 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 quasi-extremisers 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/0000-0001-5537-9693
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