论文标题
Kakeya猜想的不同形式与Hausdorff的双重性与添加剂补充的包装维度之间的等价
Equivalences between different forms of the Kakeya conjecture and duality of Hausdorff and packing dimensions for additive complements
论文作者
论文摘要
Kakeya的猜想通常被表达为以下陈述：$ {\ mathbb r}^n $的每个紧凑型/borel/tunix subset in tenection in chausdorff dimension $ n $;或者，有时，$ {\ mathbb r}^n $的每个关闭/borel/borel/intunary子集都包含各个方向的完整行都有hausdorff dimension $ n $。通常预计这些陈述是等效的。此外，该集合在各个方向上包含一条线（段）的条件通常是通过仅需要一组“大”方向的线（段）来放松的，其中大可能意味着一组正$（n-1）$ - 尺寸 - 尺寸的lebesgue度量。 在这里，我们证明了Kakeya猜想的所有上述形式确实是等效的。实际上，我们证明存在$ d \ le n $和一个紧凑的子集$ c $ $ {\ mathbb r}^n $的hausdorff dimension $ d $，其中包含每个方向沿每个方向的单位线段（以及每个方向上包含一条$ d forenderion $ d finection $ d $ s $ s $ s $ n n Divemension $ d $的nive $ d $），该$ s $ s}^\ n $ s} Hausdorff尺寸$ n-1 $，必须具有至少$ d $的尺寸。 我们还通过添加剂补充获得了Hausdorff和包装维度的二元性的结果：对于任何非空的Borel设置$ a $ a $ a $ {\ MATHBB r}^n $，我们表明 （1）$ a $的hausdorff尺寸可以作为$ n-p $获得，其中$ p $是这些borel子集的包装维度的最小值$ b $ of $ {\ mathbb r}^n $，其中$ a+a+a+a+b = {\ mathbb r}^n $;和 （2）$ a $的包装尺寸可以作为$ n-h $获得，其中$ h $是这些borel子集的hausdorff尺寸的最小值$ b $ of $ {\ mathbb r}^n $，其中$ a+a+a+b = {\ mathbb r}^n $。
The Kakeya conjecture is generally formulated as one the following statements: every compact/Borel/arbitrary subset of ${\mathbb R}^n$ that contains a (unit) line segment in every direction has Hausdorff dimension $n$; or, sometimes, that every closed/Borel/arbitrary subset of ${\mathbb R}^n$ that contains a full line in every direction has Hausdorff dimension $n$. These statements are generally expected to be equivalent. Moreover, the condition that the set contains a line (segment) in every direction is often relaxed by requiring a line (segment) for a "large" set of directions only, where large could mean a set of positive $(n-1)$-dimensional Lebesgue measure. Here we prove that all the above forms of the Kakeya conjecture are indeed equivalent. In fact, we prove that there exist $d\le n$ and a compact subset $C$ of ${\mathbb R}^n$ of Hausdorff dimension $d$ that contains a unit line segment in every direction (and also a closed set of dimension $d$ that contains a line in every direction) such that every subset $S$ of ${\mathbb R}^n$ that contains a line segment in every direction of a set of Hausdorff dimension $n-1$, must have dimension at least $d$. We also obtain results on the duality of Hausdorff and packing dimensions via additive complements: For any non-empty Borel set $A$ of ${\mathbb R}^n$ we show that (1) the Hausdorff dimension of $A$ can be obtained as $n-p$, where $p$ is the infimum of the packing dimension of those Borel subsets $B$ of ${\mathbb R}^n$ for which $A+B={\mathbb R}^n$; and (2) the packing dimension of $A$ can be obtained as $n-h$, where $h$ is the infimum of the Hausdorff dimension of those Borel subsets $B$ of ${\mathbb R}^n$ for which $A+B={\mathbb R}^n$.
