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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 resultwe 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 forjoints involving higher-dimensional affine subspaces, and it holds in thesetting of arbitrary fields $\mathbb{F}$. In contrast, for our second result,we work in the three-dimensional Euclidean space $\mathbb{R}^3$, and weestablish 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. Thisresult strengthens the known estimates for joints, including those countingmultiplicities. Additionally, our techniques yield significant structuralinformation on quasi-extremisers for this inequality.

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