论文标题
Kakeya从$ sl_2 $中的行中套装
Kakeya sets from lines in $SL_2$
论文作者
论文摘要
我们证明，每个Kakeya设置为$ \ MATHBB {r}^3 $，由$（a，b，0） + \ operatatorName {span}（c，d，d，1）$带有$ ad-bc = 1 $必须有Hausdorff dimension $ 3 $ 3 $;这种类型的Kakeya集称为$ SL_2 $ KAKEYA套装。 Fässler和Orponen最近也使用不同的技术证明了这一结果。我们的方法结合了尺度的诱导与$ sl_2 $ kakeya套件的特殊结构属性，其中说，当地的集合看起来像是平面曲线在特殊类型的地图上方排列的前图像，从$ \ mathbb {r}^3 $ to $ \ $ \ mathbb {r}^r}^2 $，称为扭曲投影。这将$ SL_2 $ Kakeya设置的研究减少了飞机曲线的Kakeya型问题；使用Wolff圆形最大函数的变体对后者进行分析。
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 Fässler 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.
