On various Carleson-type geometric lemmas and uniform rectifiability in metric spaces: Part 2
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909618689540096 |
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| author | Fässler, Katrin Violo, Ivan Yuri |
| author_facet | Fässler, Katrin Violo, Ivan Yuri |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_15421 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On various Carleson-type geometric lemmas and uniform rectifiability in metric spaces: Part 2 Fässler, Katrin Violo, Ivan Yuri Metric Geometry 49Q15, 43A80, 43A85 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. |
| title | On various Carleson-type geometric lemmas and uniform rectifiability in metric spaces: Part 2 |
| topic | Metric Geometry 49Q15, 43A80, 43A85 |
| url | https://arxiv.org/abs/2505.15421 |