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We also consider the supremum of these new dimensions, which turns out to be another interesting notion of fractal dimension.
We prove that among those bilipschitz invariant, monotone dimensions on the compact subsets of $\mathbb{R}^n$ that agree with the similarity dimension for the simplest self-similar sets, the modified lower dimension is the smallest and when $n=1$ the Assouad dimension is the greatest, and this latter statement is false for $n>1$. This answers a question of Rutar.
| Comments: | 26 pages, Theorems 4.3 and 5.14 have been added. Final, published version |
| Subjects: | Classical Analysis and ODEs (math.CA); Metric Geometry (math.MG) |
| MSC classes: | 28A78, 28A80, 51F30 |
| Cite as: | arXiv:2312.06456 [math.CA] |
| (or arXiv:2312.06456v3 [math.CA] for this version) | |
| https://doi.org/10.48550/arXiv.2312.06456 arXiv-issued DOI via DataCite |
|
| Journal reference: | Nonlinearity 38 (2025), 035024 |
From: Richard Balka [view email]
[v1]
Mon, 11 Dec 2023 15:45:15 UTC (23 KB)
[v2]
Wed, 20 Dec 2023 17:07:54 UTC (24 KB)
[v3]
Mon, 25 May 2026 03:08:34 UTC (25 KB)
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