Abstract
We adapt Guth's polynomial partitioning argument for the Fourier restriction problem to the context of the Kakeya problem. By writing out the induction argument as a recursive algorithm, additional multiscale geometric information is made available. To take advantage of this, we prove that directionseparated tubes satisfy a multiscale version of the polynomial Wolff axioms. Altogether, this yields improved bounds for the Kakeya maximal conjecture in Rn with n=5 or n≥7 and improved bounds for the Kakeya set conjecture for an infinite sequence of dimensions.
Original language  English 

Publisher  ArXiv 
Number of pages  43 
Publication status  Published  14 Aug 2019 
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Jonathan Hickman
 School of Mathematics  Lecturer in Mathematical Sciences
Person: Academic: Research Active (Teaching)