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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| Accesso online: | https://arxiv.org/abs/2604.15887 |
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| _version_ | 1866915942037979136 |
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| author | Takáč, Jakub |
| author_facet | Takáč, Jakub |
| contents | We show that whenever a separable subset $S$ of a complete metric space $X$ admits a $d$-dimensional weak tangent field, the set $S$ is close to being $d$-dimensional in the following sense. Whenever $μ$ is a Borel finite measure on $X$ supported on $S$, then a typical $1$-Lispchitz map (in the sense of Baire category) into a Euclidean space maps $μ$-almost all of $S$ into a set of Hausdorff dimension at most $d$. When taking $d=0$, this implies that any $1$-purely unrectifiable set is typically carried into a Hausdorff $0$-dimensional set up to a $μ$-null set. We show that the result is sharp in Euclidean spaces and, more generally, in strictly convex Banach spaces of finite dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_15887 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Perturbations of measures and sets having curves in d directions Takáč, Jakub Metric Geometry 28A75 (Primary) 26A16, 30L99 (Secondary) We show that whenever a separable subset $S$ of a complete metric space $X$ admits a $d$-dimensional weak tangent field, the set $S$ is close to being $d$-dimensional in the following sense. Whenever $μ$ is a Borel finite measure on $X$ supported on $S$, then a typical $1$-Lispchitz map (in the sense of Baire category) into a Euclidean space maps $μ$-almost all of $S$ into a set of Hausdorff dimension at most $d$. When taking $d=0$, this implies that any $1$-purely unrectifiable set is typically carried into a Hausdorff $0$-dimensional set up to a $μ$-null set. We show that the result is sharp in Euclidean spaces and, more generally, in strictly convex Banach spaces of finite dimension. |
| title | Perturbations of measures and sets having curves in d directions |
| topic | Metric Geometry 28A75 (Primary) 26A16, 30L99 (Secondary) |
| url | https://arxiv.org/abs/2604.15887 |