Ahlfors-David regular sets and bilipschitz maps

Iterating the Big--Pieces operator and larger sets

We show that if an Ahlfors-David regular set $E$ of dimension $k$ has Big Pieces of Big Pieces of Lipschitz Graphs (denoted usually by $BP(BP(LG))$), then $E\subset \tilde{E}$ where $\tilde{E}$ is Ahlfors-David regular of dimension $k$ and has Big Pieces of Lipschitz Graphs (denoted usually by $BP(LG)$. Our results are quantitative and, in fact, are proven in the setting of a metric space for any family of Ahlfors-David regular sets $\mathcal{F}$ replacing $LG$. A simple corollary is the stability of the BP operator after 2 iterations. This was previously only known in the Euclidean setting fo…

Ahlfors--David 正则集的和集

Sumsets of Ahlfors--David regular sets

本文证明有限 Ahlfors--David 正则集的和集大小具有下界，指数依赖正则参数，通过多尺度熵分解与加性组合逆结果实现，且依赖阶最优。

Steiner树的通用Ahlfors-David正则性

Universal Ahlfors--David regularity of Steiner trees

本文研究Steiner树的正则性，证明在d>2时，Steiner树集St_ε在任意点x和半径ρ内满足Ahlfors-David正则性，且常数仅依赖于d。

关于Ahlfors-David测度在几何平均误差下的最优Voronoi划分

On the optimal Voronoi partitions for Ahlfors-David measures with respect to the geometric mean error

本文研究Ahlfors-David概率测度在几何平均误差下n-最优集对应Voronoi划分的性质，证明了各划分单元的测度均以常数倍n^{-1}为上下界，并给出单元包含闭球半径的几何尺寸估计。

Ahlfors-Regular Curves In Metric Spaces

We discuss 1-Ahlfors-regular connected sets in a general metric space and prove that such sets are `flat' on most scales and in most locations. Our result is quantitative, and when combined with work of I. Hahlomaa, gives a characterization of 1-Ahlfors regular subsets of 1-Ahlfors-regular curves in metric spaces. Our result is a generalization to the metric space setting of the Analyst's (Geometric) Traveling Salesman theorems of P. Jones, K. Okikiolu, and G. David and S. Semmes, and it can be stated in terms of average Menger curvature.

直径有界等测划分的Ahlfors正则度量测度空间

Diameter bounded equal measure partitions of Ahlfors regular metric measure spaces

本文结合Feige和Schechtman的算法与David和Christ的dyadic立方体构造，提出一种适用于有限测度的Ahlfors正则度量测度空间的等测划分算法，该算法能够将高维球体划分为测度相等且直径较小的区域。

On the distance sets of AD-regular sets

I prove that if $\emptyset \neq K \subset \mathbb{R}^{2}$ is a compact $s$-Ahlfors-David regular set with $s \geq 1$, then $$\dim_{\mathrm{p}} D(K) = 1,$$ where $D(K) := \{|x - y| : x,y \in K\}$ is the distance set of $K$, and $\dim_{\mathrm{p}}$ stands for packing dimension. The same proof strategy applies to other problems of similar nature. For instance, one can show that if $\emptyset \neq K \subset \mathbb{R}^{2}$ is a compact $s$-Ahlfors-David regular set with $s \geq 1$, then there exists a point $x_{0} \in K$ such that $\dim_{\mathrm{p}} K \cdot (K - x_{0}) = 1$. Specialising to produc…

Quantitative Comparisons of Multiscale Geometric Properties

We generalize some characterizations of uniformly rectifiable (UR) sets to sets whose Hausdorff content is lower regular (and in particular, do not need to be Ahlfors regular). For example, David and Semmes showed that, given an Ahlfors $d$-regular set $E$, if we consider the set $\mathscr{B}$ of surface cubes (in the sense of Christ and David) near which $E$ does not look approximately like a union of planes, then $E$ is UR if and only if $\mathscr{B}$ satisfies a Carleson packing condition, that is, for any surface cube $R$, \[ \sum_{Q\subseteq R\atop Q\in \mathscr{B}} ({\rm diam} Q)^{d} \le…

Characterizations of rectifiable metric measure spaces

We characterize $n$-rectifiable metric measure spaces as those spaces that admit a countable Borel decomposition so that each piece has positive and finite $n$-densities and one of the following: is an $n$-dimensional Lipschitz differentiability space; has $n$-independent Alberti representations; satisfies David's condition for an $n$-dimensional chart. The key tool is an iterative grid construction which allows us to show that the image of a ball with a high density of curves from the Alberti representations under a chart map contains a large portion of a uniformly large ball and hence satisf…

An Analyst's Traveling Salesman Theorem for sets of dimension larger than one

In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane via a multiscale sum of $β$-numbers. These $β$-numbers are geometric quantities measuring how far a given set deviates from a best fitting line at each scale and location. Jones' result is a quantitative way of saying that a curve is rectifiable if and only if it has a tangent at almost every point. Moreover, computing this square sum for a curve returns the length of the curve up to multiplicative constant. K. Okikiolu extended his result from subsets of the plane to subsets of Euclidean space. G.…

Random Martingales and localization of maximal inequalities

Let $(X,d,μ)$ be a metric measure space. For $\emptyset\neq R\subseteq (0,\infty)$ consider the Hardy-Littlewood maximal operator $$ M_R f(x) \stackrel{\mathrm{def}}{=} \sup_{r \in R} \frac{1}{μ(B(x,r))} \int_{B(x,r)} |f| dμ.$$ We show that if there is an $n>1$ such that one has the "microdoubling condition" $ μ(B(x,(1+\frac{1}{n})r))\lesssim μ(B(x,r)) $ for all $x\in X$ and $r>0$, then the weak $(1,1)$ norm of $M_R$ has the following localization property: $$ \|M_R\|_{L_1(X) \to L_{1,\infty}(X)}\asymp \sup_{r>0} \|M_{R\cap [r,nr]}\|_{L_1(X) \to L_{1,\infty}(X)}. $$ An immediate consequence is…

具有受控几何的度量空间中的双Lipschitz弧

Bi-Lipschitz arcs in metric spaces with controlled geometry

该研究将双Lipschitz延拓结果推广到受控几何的完备度量测度空间，给出了实轴子集双Lipschitz嵌入可延拓至全直线的精确条件，还证明了小Assouad维数补集的开集是一致域及连续统的双Lipschitz曲线定量逼近结论。

Intrinsically quasi-isometric sections in metric spaces

This note is a contribution to large scale geometry. More precisely, we introduce the intrinsically quasi-isometric sections in metric spaces and we investigate their properties: the Ahlfors-David regularity in large scale; following Cheeger theory, it is possible to define suitable sets in order to obtain convexity and being a vector space over $\mathbb{R}$ or $\mathbb{C}$ for these sections; yet, following Cheeger's idea, we give an equivalence relation for this class of sections. Throughout the paper, we use basic mathematical tools.

Bi-Lipschitz Decomposition of Lipschitz functions into a Metric space

We prove a quantitative version of the following statement. Given a Lipschitz function f from the k-dimensional unit cube into a general metric space, one can decomposed f into a finite number of BiLipschitz functions f|_{F_i} so that the k-Hausdorff content of f([0,1]^k\setminus \cup F_i) is small. We thus generalize a theorem of P. Jones (1988) from the setting of R^d to the setting of a general metric space. This positively answers problem 11.13 in ``Fractured Fractals and Broken Dreams" by G. David and S. Semmes, or equivalently, question 9 from ``Thirty-three yes or no questions about map…

Bi-Lipschitz Pieces between Manifolds

A well-known class of questions asks the following: If $X$ and $Y$ are metric measure spaces and $f:X\rightarrow Y$ is a Lipschitz mapping whose image has positive measure, then must $f$ have large pieces on which it is bi-Lipschitz? Building on methods of David (who is not the present author) and Semmes, we answer this question in the affirmative for Lipschitz mappings between certain types of Ahlfors $s$-regular, topological $d$-manifolds. In general, these manifolds need not be bi-Lipschitz embeddable in any Euclidean space. To prove the result, we use some facts on the Gromov-Hausdorff con…

Hard Sard: Quantitative Implicit Function and Extension Theorems for Lipschitz Maps

We prove a global implicit function theorem. In particular we show that any Lipschitz map $f:\bR^n\times \bR^m\to\bR^n$ (with $n$-dim. image) can be precomposed with a bi-Lipschitz map $\bar{g}:\bR^n\times \bR^m\to \bR^n\times \bR^m$ such that $f\circ \bar{g}$ will satisfy, when we restrict to a large portion of the domain $E\subset \bR^n\times \bR^m$, that $f\circ \bar{g}$ is bi-Lipschitz in the first coordinate, and constant in the second coordinate. Geometrically speaking, the map $\bar{g}$ distorts $\bR^{n+m}$ in a controlled manner, so that the fibers of $f$ are straightened out. Furtherm…

Intrinsically Hölder sections in metric spaces

We introduce a notion of intrinsically Hölder graphs in metric spaces. Following a recent paper of Le Donne and the author, we prove some relevant results as the Ascoli-Arzelà compactness Theorem, Ahlfors-David regularity and the Extension Theorem for this class of sections. In the first part of this note, thanks to Cheeger theory, we define suitable sets in order to obtain a vector space over $\R$ or $\C,$ a convex set and an equivalence relation for intrinsically Hölder graphs. These last three properties are new also in the Lipschitz case. Throughout the paper, we use basic mathematical too…

一致可求长度量空间：Lipschitz 像、双侧弱几何引理与 Corona 分解

Uniformly rectifiable metric spaces: Lipschitz images, Bi-Lateral Weak Geometric Lemma and Corona Decompositions

本文在任意 Ahlfors 正则度量空间中证明了 Lipschitz 像大块、双侧弱几何引理与 Corona 分解的等价性，从而将一致可求长性的经典刻画推广到一般度量空间，并研究了 Reifenberg 参数化。

Corona Decompositions for Parabolic Uniformly Rectifiable Sets

We prove that parabolic uniformly rectifiable sets admit (bilateral) corona decompositions with respect to regular Lip(1,1/2) graphs. Together with our previous work, this allows us to conclude that if $Σ\subset\mathbb{R}^{n+1}$ is parabolic Ahlfors-David regular, then the following statements are equivalent. (1) $Σ$ is parabolic uniformly rectifiable. (2) $Σ$ admits a corona decomposition with respect to regular Lip(1,1/2) graphs. (3) $Σ$ admits a bilateral corona decomposition with respect to regular Lip(1,1/2) graphs. (4) $Σ$ is big pieces squared of regular Lip(1,1/2) graphs.

A remark on two notions of flatness for sets in the Euclidean space

In this note we compare two ways of measuring the $n$-dimensional "flatness" of a set $S\subset \mathbb{R}^d$, where $n\in \mathbb{N}$ and $d>n$. The first one is to consider the classical Reifenberg-flat numbers $α(x,r)$ ($x \in S$, $r>0$), which measure the minimal scaling-invariant Hausdorff distances in $B_r(x)$ between $S$ and $n$-dimensional affine subspaces of $\mathbb{R}^d$. The second is an `intrinsic' approach in which we view the same set $S$ as a metric space (endowed with the induced Euclidean distance). Then we consider numbers ${\sf a}(x,r)$'s, that are the scaling-invariant Gro…

Fourier integral operators, fractal sets and the regular value theorem

We prove that if ${\mathcal E} \subset {\Bbb R}^{2d}$, $d \ge 2$, is an Ahlfors-David regular product set of sufficiently large Hausdorff dimension, denoted by $dim_{\mathcal H}({\mathcal E})$, and $ϕ$ is a sufficiently regular function, then the upper Minkowski dimension of the set $$ \{w \in {\mathcal E}: ϕ_l(w)=t_l; 1 \leq l \leq m \}$$ does not exceed $dim_{\mathcal H}({\mathcal E})-m$, in line with the regular value theorem from the elementary differential geometry. Our arguments are based on the mapping properties of the underlying Fourier Integral Operators and are intimately connected …