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On various Carleson-type geometric lemmas and uniform rectifiability in metric spaces: Part 2

Katrin Fässler, Ivan Yuri Violo

TL;DR

The paper advances the theory of ι-numbers, a new class of flatness coefficients that quantify how well a set approximates model spaces through mappings rather than metric distances. It proves a Carleson-type geometric lemma for ι-numbers yields a full characterization of uniform k-rectifiability in Euclidean spaces, and it provides an abstract transfer mechanism from β-number based lemmas to ι-number lemmas that applies to general metric spaces. The second major contribution applies the abstract result to low-dimensional subsets of Heisenberg groups, establishing a Heisenberg tilting estimate and showing that β-lemmas imply ι-lemmas in this non-Euclidean setting. This work unifies and extends quantitative rectifiability theory to abstract metric spaces and to sub-Riemannian geometry, enabling robust characterizations beyond the classical Euclidean framework.

Abstract

We characterize uniform $k$-rectifiability in Euclidean spaces in terms of a Carleson-type geometric lemma for a new notion of flatness coefficients, which we call $ι$-numbers. The characterization follows from an abstract statement about approximation by generalized planes in metric spaces, which also applies to the study of low-dimensional sets in Heisenberg groups. A key aspect is that the $ι$-coefficients are in general not pointwise comparable to the usual squared $β$-numbers for dyadic cubes on $k$-regular sets in $\mathbb{R}^n$, however our result implies that they are still equivalent in terms of a Carleson-type geometric lemma.

On various Carleson-type geometric lemmas and uniform rectifiability in metric spaces: Part 2

TL;DR

The paper advances the theory of ι-numbers, a new class of flatness coefficients that quantify how well a set approximates model spaces through mappings rather than metric distances. It proves a Carleson-type geometric lemma for ι-numbers yields a full characterization of uniform k-rectifiability in Euclidean spaces, and it provides an abstract transfer mechanism from β-number based lemmas to ι-number lemmas that applies to general metric spaces. The second major contribution applies the abstract result to low-dimensional subsets of Heisenberg groups, establishing a Heisenberg tilting estimate and showing that β-lemmas imply ι-lemmas in this non-Euclidean setting. This work unifies and extends quantitative rectifiability theory to abstract metric spaces and to sub-Riemannian geometry, enabling robust characterizations beyond the classical Euclidean framework.

Abstract

We characterize uniform -rectifiability in Euclidean spaces in terms of a Carleson-type geometric lemma for a new notion of flatness coefficients, which we call -numbers. The characterization follows from an abstract statement about approximation by generalized planes in metric spaces, which also applies to the study of low-dimensional sets in Heisenberg groups. A key aspect is that the -coefficients are in general not pointwise comparable to the usual squared -numbers for dyadic cubes on -regular sets in , however our result implies that they are still equivalent in terms of a Carleson-type geometric lemma.

Paper Structure

This paper contains 20 sections, 26 theorems, 191 equations, 1 figure.

Table of Contents

  1. Introduction
  2. From $\beta$-numbers to $\iota$-numbers in Euclidean spaces
  3. Geometric lemmas in an axiomatic setting: from $\iota$- to $\beta$-numbers
  4. Preliminaries
  5. Standard quantitative notions
  6. Ahlfors regular sets and dyadic systems
  7. Geometric lemmas for various coefficient functions
  8. Relations between geometric lemmas for beta- and iota-numbers
  9. The specific case of the Euclidean space
  10. Converse inequalities in the Euclidean space
  11. Vanishing $\iota$-numbers
  12. Low-dimensional uniformly rectifiable subsets of Heisenberg groups
  13. The Heisenberg group
  14. Isotropic subspaces
  15. Horizontal projections onto affine horizontal planes
  16. ...and 5 more sections

Key Result

Theorem 1.4

A $k$-regular set $E\subset\mathbb{R}^n$ is uniformly $k$-rectifiable if and only if $E\in \mathrm{GLem}(\iota_{1,\mathcal{V}_k},1)$.

Figures (1)

  • Figure 1: Example of set $E$ where $\iota_{1,\mathcal{V}_k}\lesssim(\beta_{2,\mathcal{V}_k})^2$ fails at scale $r$.

Theorems & Definitions (57)

  • Theorem 1.4
  • Remark 1.6
  • Theorem 1.8: GLem for $\beta$-numbers implies GLem for $\iota$-numbers
  • Definition 2.1: $s$-regular sets
  • Definition 2.10: Geometric lemma
  • Definition 3.1: System of planes-projections-angle
  • Definition 3.3: $\beta$-numbers
  • Definition 3.5: $\iota$-numbers
  • Remark 3.7
  • Proposition 3.8
  • ...and 47 more