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

TL;DR

This paper establishes sharp bounds on the number of joints formed by lines and k-planes over arbitrary fields, and provides a Kakeya-type estimate in Euclidean space, advancing understanding of geometric incidences and extremisers.

Contribution

It introduces the first sharp bounds for joints involving higher-dimensional affine subspaces over arbitrary fields and proves a new Kakeya-type estimate in Euclidean space, with structural insights.

Findings

Sharp bound on joints formed by k-planes and lines in arbitrary fields

Kakeya-type estimate in R^3 for joints with multiplicity considerations

Structural information on quasi-extremisers for the Kakeya inequality

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 $O_n(|\mathcal{L}||\mathcal{P}|^{1/(n-k)}$). 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 \begin{equation*}\sum_{x \in J} \left(\sum_{\ell \in \mathcal{L}} \chi_\ell(x)\right)^{3/2} \lesssim |\mathcal{L}|^{3/2}\end{equation*} 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.