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Restricting directions for Kakeya sets

Authors
  • Gauvan, Anthony
Publication Date
Apr 04, 2022
Source
HAL
Keywords
Language
English
License
Unknown
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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 have dX ≥ 6 11 dΩ + 1 where dE denotes the Hausdorff dimension of a set E.

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