The weak lower density condition and uniform rectifiability

We show that an Ahlfors $d$-regular set $E$ in $\mathbb{R}^{n}$ is uniformly rectifiable if the set of pairs $(x,r)\in E\times (0,\infty)$ for which there exists $y \in B(x,r)$ and $01+\epsilon\right\} \le_{C,A,\epsilon,\rho,s} \mathbf{H}^s(X)$. This is a quantitative version of the cl...

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Veröffentlicht in:	Annales Fennici Mathematici 2022-01, Vol.47 (2), p.791-819
Hauptverfasser:	Azzam, Jonas, Hyde, Matthew
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creator	Azzam, Jonas Hyde, Matthew
description	We show that an Ahlfors $d$-regular set $E$ in $\mathbb{R}^{n}$ is uniformly rectifiable if the set of pairs $(x,r)\in E\times (0,\infty)$ for which there exists $y \in B(x,r)$ and $01+\epsilon\right\} \le_{C,A,\epsilon,\rho,s} \mathbf{H}^s(X)$. This is a quantitative version of the classical result that for a metric space $X$ of finite $s$-dimensional Hausdorff measure, the upper $s$-dimensional densities are at most 1 $\mathbf{H}^s$-almost everywhere.
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To prove this, we generalize a result of Schul by proving, if $X$ is a $C$-doubling metric space, $\epsilon,\rho\in (0,1)$, $A>1$, and $X_n$ is a sequence of maximal $2^{-n}$-separated sets in $X$, and $\mathbf{B}=\{B(x,2^{-n})\colon x\in X_{n},n\in \mathbb{N}\}$, then $\sum \left\{r_{B}^s\colon B\in \mathbf{B}, \frac{\mathbf{H}^s_{\rho r_{B}}(X\cap AB)}{(2r_{AB})^s}>1+\epsilon\right\} \le_{C,A,\epsilon,\rho,s} \mathbf{H}^s(X)$. This is a quantitative version of the classical result that for a metric space $X$ of finite $s$-dimensional Hausdorff measure, the upper $s$-dimensional densities are at most 1 $\mathbf{H}^s$-almost everywhere. </description><identifier>ISSN: 2737-0690</identifier><identifier>EISSN: 2737-114X</identifier><identifier>DOI: 10.54330/afm.119478</identifier><language>eng</language><ispartof>Annales Fennici Mathematici, 2022-01, Vol.47 (2), p.791-819</ispartof><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c270t-2ad8e998e7a5198dca8dddcfd23b2ae75c7a3951afac4530e04177d930dd07dc3</citedby></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,778,782,27911,27912</link.rule.ids></links><search><creatorcontrib>Azzam, Jonas</creatorcontrib><creatorcontrib>Hyde, Matthew</creatorcontrib><title>The weak lower density condition and uniform rectifiability</title><title>Annales Fennici Mathematici</title><description> We show that an Ahlfors $d$-regular set $E$ in $\mathbb{R}^{n}$ is uniformly rectifiable if the set of pairs $(x,r)\in E\times (0,\infty)$ for which there exists $y \in B(x,r)$ and $0<t<r$ satisfying $\mathbf{H}^{d}_{\infty}(E\cap B(y,t))<(2t)^{d}-\epsilon(2r)^d$ is a Carleson set for every $\epsilon>0$. To prove this, we generalize a result of Schul by proving, if $X$ is a $C$-doubling metric space, $\epsilon,\rho\in (0,1)$, $A>1$, and $X_n$ is a sequence of maximal $2^{-n}$-separated sets in $X$, and $\mathbf{B}=\{B(x,2^{-n})\colon x\in X_{n},n\in \mathbb{N}\}$, then $\sum \left\{r_{B}^s\colon B\in \mathbf{B}, \frac{\mathbf{H}^s_{\rho r_{B}}(X\cap AB)}{(2r_{AB})^s}>1+\epsilon\right\} \le_{C,A,\epsilon,\rho,s} \mathbf{H}^s(X)$. 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To prove this, we generalize a result of Schul by proving, if $X$ is a $C$-doubling metric space, $\epsilon,\rho\in (0,1)$, $A>1$, and $X_n$ is a sequence of maximal $2^{-n}$-separated sets in $X$, and $\mathbf{B}=\{B(x,2^{-n})\colon x\in X_{n},n\in \mathbb{N}\}$, then $\sum \left\{r_{B}^s\colon B\in \mathbf{B}, \frac{\mathbf{H}^s_{\rho r_{B}}(X\cap AB)}{(2r_{AB})^s}>1+\epsilon\right\} \le_{C,A,\epsilon,\rho,s} \mathbf{H}^s(X)$. This is a quantitative version of the classical result that for a metric space $X$ of finite $s$-dimensional Hausdorff measure, the upper $s$-dimensional densities are at most 1 $\mathbf{H}^s$-almost everywhere. </abstract><doi>10.54330/afm.119478</doi><tpages>29</tpages><oa>free_for_read</oa></addata></record>
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title	The weak lower density condition and uniform rectifiability
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