Attainable forms of Assouad spectra
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866912207687647232 |
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| author | Rutar, Alex |
| author_facet | Rutar, Alex |
| contents | Let $d\in\mathbb{N}$ and let $φ\colon(0,1)\to[0,d]$. We prove that there exists a set $F\subset\mathbb{R}^d$ such that $\operatorname{dim}_A^θF=φ(θ)$ for all $θ\in(0,1)$ if and only if for every $0<λ<θ<1$, \[0\leq (1-λ)φ(λ)-(1-θ)φ(θ)\leq (θ-λ)φ\Bigl(\fracλθ\Bigr).\] In particular, the following behaviours which have not previously been witnessed in any examples are possible: the Assouad spectrum can be non-monotonic on every open set, and can fail to be Hölder in a neighbourhood of 1. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_06921 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Attainable forms of Assouad spectra Rutar, Alex Classical Analysis and ODEs Dynamical Systems Metric Geometry 28A80 (Primary) 39B62 (Secondary) Let $d\in\mathbb{N}$ and let $φ\colon(0,1)\to[0,d]$. We prove that there exists a set $F\subset\mathbb{R}^d$ such that $\operatorname{dim}_A^θF=φ(θ)$ for all $θ\in(0,1)$ if and only if for every $0<λ<θ<1$, \[0\leq (1-λ)φ(λ)-(1-θ)φ(θ)\leq (θ-λ)φ\Bigl(\fracλθ\Bigr).\] In particular, the following behaviours which have not previously been witnessed in any examples are possible: the Assouad spectrum can be non-monotonic on every open set, and can fail to be Hölder in a neighbourhood of 1. |
| title | Attainable forms of Assouad spectra |
| topic | Classical Analysis and ODEs Dynamical Systems Metric Geometry 28A80 (Primary) 39B62 (Secondary) |
| url | https://arxiv.org/abs/2206.06921 |