Kakeya sets from lines in $S{L}_2$

Abstract

We prove that every Kakeya set in ${\mathbb{R}}^3$ formed from lines of the form $(a,b,0)+\mathrm{span}(c,d,1)$ with $ad-bc=1$ must have Hausdorff dimension $3$; Kakeya sets of this type are called $S{L}_2$ Kakeya sets. This result was also recently proved by Fässler and Orponen using different techniques. Our method combines induction on scales with a special structural property of $S{L}_2$ Kakeya sets, which says that locally such sets look like the pre-image of an arrangement of plane curves above a special type of map from ${\mathbb{R}}^3$ to ${\mathbb{R}}^2$, called a twisting projection. This reduces the study of $S{L}_2$ Kakeya sets to a Kakeya-type problem for plane curves; the latter is analyzed using a variant of Wolff's circular maximal function.

23 pages, 1 figure. v2: Final version, published by Ars Inveniendi Analytica