2022/04/04 by Gauvan, Anthony
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2204.01408
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.