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

Voir la notice de l'article provenant de la source Ars Inveniendi Analytica website

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 
@article{10_15781_ep89_2217,
     author = {Nets Hawk Katz and Shukun Wu and Joshua Zahl},
     title = {Kakeya sets from lines in $SL_2$},
     journal = {Ars Inveniendi Analytica},
     publisher = {mathdoc},
     year = {2023},
     doi = {10.15781/ep89-2217},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.15781/ep89-2217/}
}
TY  - JOUR
AU  - Nets Hawk Katz
AU  - Shukun Wu
AU  - Joshua Zahl
TI  - Kakeya sets from lines in $SL_2$
JO  - Ars Inveniendi Analytica
PY  - 2023
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.15781/ep89-2217/
DO  - 10.15781/ep89-2217
LA  - en
ID  - 10_15781_ep89_2217
ER  - 
%0 Journal Article
%A Nets Hawk Katz
%A Shukun Wu
%A Joshua Zahl
%T Kakeya sets from lines in $SL_2$
%J Ars Inveniendi Analytica
%D 2023
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.15781/ep89-2217/
%R 10.15781/ep89-2217
%G en
%F 10_15781_ep89_2217
Nets Hawk Katz; Shukun Wu; Joshua Zahl. Kakeya sets from lines in $SL_2$. Ars Inveniendi Analytica (2023). doi: 10.15781/ep89-2217

Cité par Sources :