Geodesic
Kakeya sets from lines in $SL_2$
Ars Inveniendi Analytica (2023).

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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.
Publié le :
DOI : 10.15781/ep89-2217 
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     author = {Nets Hawk Katz and Shukun Wu and Joshua Zahl},
     title = {Kakeya sets from lines in $SL_2$},
     journal = {Ars Inveniendi Analytica},
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     year = {2023},
     doi = {10.15781/ep89-2217},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.15781/ep89-2217/}
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Nets Hawk Katz; Shukun Wu; Joshua Zahl. Kakeya sets from lines in $SL_2$. Ars Inveniendi Analytica (2023). doi : 10.15781/ep89-2217. http://geodesic.mathdoc.fr/articles/10.15781/ep89-2217/

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