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

TL;DR

This paper characterizes uniform rectifiability in Euclidean and metric spaces using new flatness coefficients called $$-numbers, extending geometric lemmas to more general settings like Heisenberg groups.

Contribution

It introduces $$-numbers as a novel notion of flatness and establishes their equivalence to classical $eta$-numbers via Carleson-type lemmas, broadening the understanding of rectifiability.

Findings

$$-numbers provide a new geometric measure of flatness.

The characterization applies to Euclidean spaces and Heisenberg groups.

$$-coefficients are equivalent to $eta$-numbers in a Carleson sense.

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 $\iota$-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 $\iota$-coefficients are in general not pointwise comparable to the usual squared $\beta$-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.