Improved bounds for the Kakeya maximal conjecture in higher dimensions

Jonathan Hickman, Keith M. Rogers, Ruixiang Zhang

Research output: Contribution to journalArticlepeer-review

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 direction-separated tubes satisfy a multiscale version of the polynomial Wolff axioms. Altogether, this yields improved bounds for the Kakeya maximal conjecture in $\mathbb{R}^n$ with $n=5$ or $n\ge 7$ and improved bounds for the Kakeya set conjecture for an infinite sequence of dimensions.
Original languageEnglish
Number of pages43
JournalAmerican Journal of Mathematics
Publication statusAccepted/In press - 24 Jun 2020

Keywords

  • math.CA
  • math.MG
  • 28A78, 42B99

Fingerprint Dive into the research topics of 'Improved bounds for the Kakeya maximal conjecture in higher dimensions'. Together they form a unique fingerprint.

Cite this