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arXiv · 2211.05194

Kakeya sets from lines in $SL_2$

Nets Hawk Katz·Shukun Wu·Joshua Zahl
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Abstract

We prove that every Kakeya set in $\mathbb{R}^3$ formed from lines of the form $(a,b,0) + \operatorname{span}(c,d,1)$ with $ad-bc=1$ must have Hausdorff dimension $3$; Kakeya sets of this type are called $SL_2$ Kakeya sets. This result was also recently proved by F\"assler and Orponen using different techniques. Our method combines induction on scales with a special structural property of $SL_2$ 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 $SL_2$ Kakeya sets to a Kakeya-type problem for plane curves; the latter is analyzed using a variant of Wolff's circular maximal function.

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Nets Hawk Katz, Shukun Wu, Joshua Zahl. 2022-11-09. Kakeya sets from lines in $SL_2$. https://doi.org/10.15781/ep89-2217

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