论文标题
a $ d $维分析师的旅行推销员定理，用于$ \ mathbb {r}^n $
A $d$-dimensional Analyst's Travelling Salesman Theorem for general sets in $\mathbb{R}^n$
论文作者
论文摘要
琼斯在他的1990年论文中证明了以下内容：给定$ e \ subseteq \ mathbb {r}^2 $，存在曲线$γ$，使得$ e \subseteqγ$和\ [\ mathscr {h} β_{e}（3q）^2 \ ell（q）。\ \]，$β_e（q）$测量$ e $从$ q $内的直线偏离了多远。 Okikiolu将其扩展到$ \ Mathbb {r}^n $的子集，并将其扩展到Schul到Hilbert Space的子集。 2018年，阿扎姆（Azzam）和舒尔（Schul）推出了琼斯$β$ number的一种变体。这样，他们和分别是别墅，证明了$ \ mathbb {r}^n。$的较低常规子集的结果，尤其是，别墅证明了，给定$ e \ subseteq \ subseteq \ mathbb {r}^n $（较低的内容）正常的内容较低，有一个nice $ d $ d $ d $ d $ d $ d $ f $ f $ e \ suirign $ sequeq。 \ Mathscr {h}^d（f）\ sim \ text {diam}（e）^d + \ sum_ {q}β_{e}（3q）（3Q）^2 \ ell（q）^d。 \ end {align}在此上下文中，如果满足某些拓扑非退化条件，则集合$ f $是“ nice”，首先在2004年的David论文中引入。 在本文中，我们放弃了较低的规律性条件，并证明了一般$ d $二维的子集的$ \ mathbb {r}^n。$为此，我们为此介绍了一个新的$ d $ d $二维变体，用于$β$ -number的任何定义，该变体定义了$ \ mathbbbbbb {r}^n的任何设置。
In his 1990 paper, Jones proved the following: given $E \subseteq \mathbb{R}^2$, there exists a curve $Γ$ such that $E \subseteq Γ$ and \[ \mathscr{H}^1(Γ) \sim \text{diam}\, E + \sum_{Q} β_{E}(3Q)^2\ell(Q).\] Here, $β_E(Q)$ measures how far $E$ deviates from a straight line inside $Q$. This was extended by Okikiolu to subsets of $\mathbb{R}^n$ and by Schul to subsets of a Hilbert space. In 2018, Azzam and Schul introduced a variant of the Jones $β$-number. With this, they, and separately Villa, proved similar results for lower regular subsets of $\mathbb{R}^n.$ In particular, Villa proved that, given $E \subseteq \mathbb{R}^n$ which is lower content regular, there exists a `nice' $d$-dimensional surface $F$ such that $E \subseteq F$ and \begin{align} \mathscr{H}^d(F) \sim \text{diam}( E)^d + \sum_{Q} β_{E}(3Q)^2\ell(Q)^d. \end{align} In this context, a set $F$ is `nice' if it satisfies a certain topological non degeneracy condition, first introduced in a 2004 paper of David. In this paper we drop the lower regularity condition and prove an analogous result for general $d$-dimensional subsets of $\mathbb{R}^n.$ To do this, we introduce a new $d$-dimensional variant of the Jones $β$-number that is defined for any set in $\mathbb{R}^n.$
