Restricting directions for Kakeya sets

Anthony Gauvan (1, 2)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 LMO - Laboratoire de Mathématiques d'Orsay

We prove that the Kakeya maximal conjecture is equivalent to the Ω-Kakeya maximal conjecture. This completes a recent result in [2] where Keleti and Mathé proved that the Kakeya conjecture is equivalent to the Ω-Kakeya conjecture. Moreover, we improve concrete bound on the Hausdorff dimension of a Ω-Kakeya set : for any Bore set Ω in S n−1 , we prove that if X ⊂ R n contains for any e ∈ Ω a unit segment oriented along e then we have dX ≥ 6 11 dΩ + 1 where dE denotes the Hausdorff dimension of a set E.

Domaines: Analyse classique [math.CA]; Géométrie métrique [math.MG]

Anthony Gauvan. Restricting directions for Kakeya sets. 2022. ⟨hal-03628966⟩