New curved Kakeya estimates

Abstract

We prove that for an open class of translation-invariant Hörmander-type phase functions, the corresponding curved Kakeya sets in ${\mathbb{R}}^3$ have Hausdorff dimension at least $(13-\sqrt{13})/4=2.348\dots$. Previous methods, such as polynomial partitioning, have attained at best dimension $2+1/3$ for curved Kakeya sets in ${\mathbb{R}}^3$. The proof combines Wolff's classical hairbrush argument with a new Kakeya estimate for 3-parameter families of curves satisfying stable geometric conditions that we call coniness and twistiness. The latter estimate extends ideas of Katz, Wu, and Zahl from the study of ${\mathrm{SL}}_2$-Kakeya sets.

53 pages, 3 figures. Revised after referee reports. Introduction substantially streamlined and minor errors corrected throughout. Abstract made more precise. Previous version contains a more extensive introductory discussion