Quasisymmetric maps on Kakeya sets

TL;DR

This paper links $L^{p}-L^{q}$ estimates for the Kakeya maximal function to bounds on the conformal and Hausdorff dimensions of Kakeya sets, revealing limitations on quasisymmetries' effects on these dimensions.

Contribution

It establishes new lower bounds for the conformal dimension of Kakeya sets and upper bounds for the dimension increase of line segments under quasisymmetries, connecting harmonic analysis and geometric measure theory.

Findings

Conformal dimension of Kakeya sets in $\

Bounds on the dimension of images of line segments under quasisymmetries.

Implications of the Kakeya maximal function conjecture for dimension bounds.

Abstract

I show that $L^{p}-L^{q}$ estimates for the Kakeya maximal function yield lower bounds for the conformal dimension of Kakeya sets, and upper bounds for how much quasisymmetries can increase the Hausdorff dimension of line segments inside Kakeya sets. Combining the known $L^{p}-L^{q}$ estimates of Wolff and Katz-Tao with the main result of the paper, the conformal dimension of Kakeya sets in $\mathbb{R}^{n}$ is at least $\max\{(n + 2)/2,(4n + 3)/7\}$. Moreover, if $f$ is a quasisymmetry from a Kakeya set $K \subset \mathbb{R}^{n}$ onto any at most $n$-dimensional metric space, the $f$-image of a.e. line segment inside $K$ has dimension at most $\min\{2n/(n + 2),7n/(4n + 3)\}$. The Kakeya maximal function conjecture implies that the bounds can be improved to $n$ and $1$, respectively.