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)
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Official URL: https://discreteanalysisjournal.com/article/18361-... |
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 |
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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: | 04 Mar 2024 18:33 UTC |
Resource URI: | https://kar.kent.ac.uk/id/eprint/85214 (The current URI for this page, for reference purposes) |
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