arXiv subjects

Raanan Schul

Publications and source records attributed to Raanan Schul.

At least 19 recordsLinked to original sources

arXiv2025

Characterizing rectifiability via biLipschitz pieces of Lipschitz mappings on the space

We give the following characterization of rectifiable metric spaces. A metric space with positive lower Hausdorff density is rectifiable if and only if, for any subset $F$ and $f:F\to Y$, a Lipschitz map into a metric space with positive measure image (of the same dimension), there exists a positive measure subset $A\subset F$ so that $f$ is biLipschitz on $A$. We also give a characterization in terms of a full biLipschitz decomposition. These characterizations are new even for subsets of Euclidean space. One of our tools is Alberti representations. On the way we give a method for constructing independent Alberti representations, which may be of independent interest. We use this to characterize unrectifiable metric spaces as those spaces for which there exist a positive measure subset $S$ and a Lipschitz map $\phi$ into a lower dimensional Euclidean space so that $S$ is $\cH^1$-null with respect to all curve fragments that are quantitatively transversal to $\phi$.

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arXiv2025

Coarse and pointwise tangent fields

Alberti, Cs\"ornyei and Preiss introduced a notion of a "pointwise (weak) tangent field" for a subset of Euclidean space -- a field that contains almost every tangent line of every curve passing through the set -- and showed that all area-zero sets in the plane admit one-dimensional tangent fields. We extend their results in two distinct directions. First, a special case of our pointwise result shows that each doubling subset of Hilbert space admits a pointwise tangent field in this sense, with dimension bounded by the Nagata (or Assouad) dimension of the set. Second, inspired by the Analyst's Traveling Salesman Theorem of Jones, we introduce new, "coarse" notions of tangent field for subsets of Hilbert space, which take into account both large and small scale structure. We show that doubling subsets of Hilbert space admit such coarse tangent fields, again with dimension bounded by the Nagata (or Assouad) dimension of the set. For porous sets in the plane, this result can be viewed as a quantitative version of the Alberti--Cs\"ornyei--Preiss result, though our results hold in all (even infinite) dimensions.

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arXiv2024

Factorization and piecewise affine approximation of bi-Lipschitz mappings on large sets

A well-known open problem asks whether every bi-Lipschitz homeomorphism of $\mathbb{R}^d$ factors as a composition of mappings of small distortion. We show that every bi-Lipschitz embedding of the unit cube $[0,1]^d$ into $\mathbb{R}^d$ factors into finitely many global bi-Lipschitz mappings of small distortion, outside of an exceptional set of arbitrarily small Lebesgue measure, which cannot in general be removed. Our main tool is a corona-type decomposition theorem for bi-Lipschitz mappings. As corollaries, we obtain a related factorization result for bi-Lipschitz homeomorphisms of the $d$-sphere, and we show that bi-Lipschitz embeddings of the unit $d$-cube in $\mathbb{R}^d$ can be approximated by global piecewise affine homeomorphisms outside of a small set.

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arXiv2023

Square packings and rectifiable doubling measures

We prove that for all integers $2\leq m\leq d-1$, there exists doubling measures on $\mathbb{R}^d$ with full support that are $m$-rectifiable and purely $(m-1)$-unrectifiable in the sense of Federer (i.e. without assuming $\mu\ll\mathcal{H}^m$). The corresponding result for 1-rectifiable measures is originally due to Garnett, Killip, and Schul (2010). Our construction of higher-dimensional Lipschitz images is informed by a simple observation about square packing in the plane: $N$ axis-parallel squares of side length $s$ pack inside of a square of side length $\lceil N^{1/2}\rceil s$. The approach is robust and when combined with standard metric geometry techniques allows for constructions in complete Ahlfors regular metric spaces. One consequence of the main theorem is that for each $m\in\{2,3,4\}$ and $s 0$, $f(E)$ has Hausdorff dimension $s$, and $\mu(f(E))>0$. This is striking, because $\mathcal{H}^m(f(E))=0$ for every Lipschitz map $f:E\subset\mathbb{R}^m\rightarrow\mathbb{H}^1$ by a theorem of Ambrosio and Kirchheim (2000). Another application of the square packing construction is that every compact metric space $\mathbb{X}$ of Assouad dimension strictly less than $m$ is a Lipschitz image of a compact set $E\subset[0,1]^m$. Of independent interest, we record the existence of doubling measures on complete Ahlfors regular metric spaces with prescribed lower and upper Hausdorff and packing dimensions.

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arXiv2023

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

In their 1991 and 1993 foundational monographs, David and Semmes characterized uniform rectifiability for subsets of Euclidean space in a multitude of geometric and analytic ways. The fundamental geometric conditions can be naturally stated in any metric space and it has long been a question of how these concepts are related in this general setting. In this paper we prove their equivalence. Namely, we show the equivalence of Big Pieces of Lipschitz Images, Bi-lateral Weak Geometric Lemma and Corona Decomposition in any Ahlfors regular metric space. Loosely speaking, this gives a quantitative equivalence between having Lipschitz charts and approximations by nicer spaces. En route, we also study Reifenberg parameterizations.

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arXiv2021

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 for the case $\mathcal{F}= LG$ with substantially more complicated proofs.

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arXiv2021

Lower bounds on mapping content and quantitative factorization through trees

We give a simple quantitative condition, involving the "mapping content" of Azzam--Schul, that implies that a Lipschitz map from a Euclidean space to a metric space must be close to factoring through a tree. Using results of Azzam--Schul and the present authors, this gives simple checkable conditions for a Lipschitz map to have a large piece of its domain on which it behaves like an orthogonal projection. The proof involves new lower bounds and continuity statements for mapping content, and relies on a "qualitative" version of the main theorem recently proven by Esmayli--Haj{\l}asz.

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arXiv2020

Quantitative decompositions of Lipschitz mappings into metric spaces

We study the quantitative properties of Lipschitz mappings from Euclidean spaces into metric spaces. We prove that it is always possible to decompose the domain of such a mapping into pieces on which the mapping "behaves like a projection mapping" along with a "garbage set" that is arbitrarily small in an appropriate sense. Moreover, our control is quantitative, i.e., independent of both the particular mapping and the metric space it maps into. This improves a theorem of Azzam-Schul from the paper "Hard Sard", and answers a question left open in that paper. The proof uses ideas of quantitative differentiation, as well as a detailed study of how to supplement Lipschitz mappings by additional coordinates to form bi-Lipschitz mappings.

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arXiv2019

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. David and S. Semmes extended the discussion to include sets of (integer) dimension larger than one, under the assumption of Ahlfors regularity and using a variant of Jones' $β$ numbers. In this paper we give a version of P. Jones' theorem for sets of arbitrary (integer) dimension lying in Euclidean space. We estimate the $d$-dimensional Hausdorff measure of a set in terms of an analogous sum of $β$-type numbers. There is no assumption of Ahlfors regularity, but rather, only of a lower bound on the Hausdorff content. We adapt David and Semmes' version of Jones' $β$-numbers by redefining them using a Choquet integral. A key tool in the proof is G. David and T. Toro's parametrization of Reifenberg flat sets (with holes).

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arXiv2019

A sharp necessary condition for rectifiable curves in metric spaces

In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane, using a multiscale sum of what is now known as Jones $β$-numbers, numbers measuring flatness in a given scale and location. This work was generalized to R^n by Okikiolu, to Hilbert space by the second author, and has many variants in a variety of metric settings. Notably, in 2005, Hahlomaa gave a sufficient condition for a subset of a metric space to be contained in a rectifiable curve. We prove the sharpest possible converse to Hahlomaa's theorem for doubling curves, and then deduce some corollaries for subsets of metric and Banach spaces, as well as the Heisenberg group.}

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arXiv2016

The Analyst's traveling salesman theorem in graph inverse limits

We prove a version of Peter Jones' Analyst's traveling salesman theorem in a class of highly non-Euclidean metric spaces introduced by Laakso and generalized by Cheeger-Kleiner. These spaces are constructed as inverse limits of metric graphs, and include examples which are doubling and have a Poincare inequality. We show that a set in one of these spaces is contained in a rectifiable curve if and only if it is quantitatively "flat" at most locations and scales, where flatness is measured with respect to so-called monotone geodesics. This provides a first examination of quantitative rectifiability within these spaces.

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arXiv2016

Multiscale analysis of 1-rectifiable measures II: characterizations

A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures $\mu$ in $n$-dimensional Euclidean space for all $n\geq 2$ in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an $L^2$ gauge the extent to which $\mu$ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the classical theorems of Besicovitch, Morse and Randolph, and Moore, we do not assume an a priori relationship between $\mu$ and 1-dimensional Hausdorff measure. We also characterize purely 1-unrectifiable Radon measures, i.e. locally finite measures that give measure zero to every finite length curve. Characterizations of this form were originally conjectured to exist by P. Jones. Along the way, we develop an $L^2$ variant of P. Jones' traveling salesman construction, which is of independent interest.

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arXiv2015

An upper bound for the length of a Traveling Salesman path in the Heisenberg group

We show that a sufficient condition for a subset $E$ in the Heisenberg group (endowed with the Carnot-Carathéodory metric) to be contained in a rectifiable curve is that it satisfies a modified analogue of Peter Jones's geometric lemma. Our estimates improve on those of \cite{FFP}, by replacing the power $2$ of the Jones-$β$-number with any power $r<4$. This complements (in an open ended way) our work \cite{Li-Schul-beta-leq-length}, where we showed that such an estimate was necessary, but with $r=4$.

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arXiv2014

Two sufficient conditions for rectifiable measures

We identify two sufficient conditions for locally finite Borel measures on $\mathbb{R}^n$ to give full mass to a countable family of Lipschitz images of $\mathbb{R}^m$. The first condition, extending a prior result of Pajot, is a sufficient test in terms of $L^p$ affine approximability for a locally finite Borel measure $\mu$ on $\mathbb{R}^n$ satisfying the global regularity hypothesis $$\limsup_{r\downarrow 0} \mu(B(x,r))/r^m <\infty\quad \text{at $\mu$-a.e. $x\in\mathbb{R}^n$}$$ to be $m$-rectifiable in the sense above. The second condition is an assumption on the growth rate of the 1-density that ensures a locally finite Borel measure $\mu$ on $\mathbb{R}^n$ with $$\lim_{r\downarrow 0} \mu(B(x,r))/r=\infty\quad\text{at $\mu$-a.e. $x\in\mathbb{R}^n$}$$ is 1-rectifiable.

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arXiv2014

Multiscale analysis of 1-rectifiable measures: necessary conditions

We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in $\Bbb{R}^n$, $n\geq 2$. To each locally finite Borel measure $μ$, we associate a function $\widetilde J_2(μ, x)$ which uses a weighted sum to record how closely the mass of $μ$ is concentrated on a line in the triples of dyadic cubes containing $x$. We show that $\widetilde J_2(μ, x) < \infty$ $μ$-a.e. is a necessary condition for $μ$ to give full mass to a countable family of rectifiable curves. This confirms a conjecture of Peter Jones from 2000. A novelty of this result is that no assumption is made on the upper Hausdorff density of the measure. Thus we are able to analyze generic 1-rectifiable measures that are mutually singular with the 1-dimensional Hausdorff measure.

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arXiv2014

The traveling salesman problem in the Heisenberg group: upper bounding curvature

We show that if a subset $K$ in the Heisenberg group (endowed with the Carnot-Carathéodory metric) is contained in a rectifiable curve, then it satisfies a modified analogue of Peter Jones's geometric lemma. This is a quantitative version of the statement that a finite length curve has a tangent at almost every point. This condition complements that of \cite{FFP} except a power 2 is changed to a power 4. Two key tools that we use in the proof are a geometric martingale argument like that of \cite{Schul-TSP} as well as a new curvature inequality in the Heisenberg group.

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arXiv2012

A quantitative metric differentiation theorem

The purpose of this note is to point out a simple consequence of some earlier work of the authors, "Hard Sard: Quantitative implicit function and extension theorems for Lipschitz maps". For $f$, a Lipschitz function from a Euclidean space into a metric space, we give quantitative estimates for how often the pullback of the metric under $f$ is approximately a seminorm. This is a quantitative version of Kirchheim's metric differentiation result from 1994. Our result is in the form of a Carleson-type estimate.

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arXiv2012

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. Furthermore, our results stay valid when the target space is replaced by {\bf any metric space}. A main point is that our results are quantitative: the size of the set $E$ on which behavior is good is a significant part of the discussion. Our estimates are motivated by examples such as Kaufman's 1979 construction of a $C^1$ map from $[0,1]^3$ onto $[0,1]^2$ with rank $\leq 1$ everywhere. On route we prove an extension theorem which is of independent interest. We show that for any $D\geq n$, any Lipschitz function $f:[0,1]^n\to \bR^D$ gives rise to a large (in an appropriate sense) subset $E\subset [0,1]^n$ such that $f|_E$ is bi-Lipschitz and may be extended to a bi-Lipschitz function defined on {\bf all} of $\bR^n$. The most interesting case is the case $D=n$. As a simple corollary, we show that $n$-dimensional Ahlfors-David regular spaces lying in $\bR^{D}$ having big pieces of bi-Lipschitz images also have big pieces of big pieces of Lipschitz graphs in $\bR^{D}$. This was previously known only for $D\geq 2n+1$ by a result of G. David and S. Semmes.

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