Restricting directions for Kakeya sets

Abstract

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 havewhere dE denotes the Hausdorff dimension of a set E.