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

Anthony Carbery, Marina Iliopoulou

Research output: Contribution to journalArticlepeer-review


Let L be a family of lines and let P be a family of k-planes in Fn where F is a field. In our first result we show that the number of joints formed by a k-plane in P together with (n−k) lines in L is On(|L||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 F. In contrast, for our second result, we work in the three-dimensional Euclidean space R3, and we establish the Kakeya-type estimate ∑x∈J(∑ℓ∈Lχℓ(x))3/2≲|L|3/2 where J is the set of joints formed by 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.
Original languageEnglish
Article number18
Pages (from-to)1-45
JournalDiscrete Analysis
Publication statusPublished - 29 Dec 2020


  • 52C99 Discrete geometry
  • 42B-99 Harmonic Analysis in Several Variables


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