Attainable forms of lower spectra
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911483079688192 |
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| author | Banaji, Amlan Chen, Haipeng Rutar, Alex Wang, Wen |
| author_facet | Banaji, Amlan Chen, Haipeng Rutar, Alex Wang, Wen |
| contents | Let $d\in\mathbb{N}$ and $φ\colon(0,1)\to[0,d]$. We prove there exists a set $F\subset\mathbb{R}^d$ whose lower spectrum $\operatorname{dim}^θ_{\mathrm{L}} F$ satisfies $(1-θ)\operatorname{dim}^θ_{\mathrm{L}} F = φ(θ)$ for all $θ\in(0,1)$ if and only if for all $λ,θ\in(0,1)$, \begin{equation*}
φ(θ) \leq φ(λθ) - θφ(λ) \leq (1-θ) d. \end{equation*} We also obtain a similar classification result for $\underline{\operatorname{dim}}^θ_{\mathrm{L}} F$.
In contrast to the case for Assouad spectra, it is insufficient to consider homogeneous (or uniform) sets. Instead, we follow the approach introduced by Orgoványi--Rutar in arXiv:2510.07013 and proceed via a more general classification result for appropriate two-scale branching functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_03174 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Attainable forms of lower spectra Banaji, Amlan Chen, Haipeng Rutar, Alex Wang, Wen Classical Analysis and ODEs Metric Geometry 28A80 (Primary) 39B62 (Secondary) Let $d\in\mathbb{N}$ and $φ\colon(0,1)\to[0,d]$. We prove there exists a set $F\subset\mathbb{R}^d$ whose lower spectrum $\operatorname{dim}^θ_{\mathrm{L}} F$ satisfies $(1-θ)\operatorname{dim}^θ_{\mathrm{L}} F = φ(θ)$ for all $θ\in(0,1)$ if and only if for all $λ,θ\in(0,1)$, \begin{equation*} φ(θ) \leq φ(λθ) - θφ(λ) \leq (1-θ) d. \end{equation*} We also obtain a similar classification result for $\underline{\operatorname{dim}}^θ_{\mathrm{L}} F$. In contrast to the case for Assouad spectra, it is insufficient to consider homogeneous (or uniform) sets. Instead, we follow the approach introduced by Orgoványi--Rutar in arXiv:2510.07013 and proceed via a more general classification result for appropriate two-scale branching functions. |
| title | Attainable forms of lower spectra |
| topic | Classical Analysis and ODEs Metric Geometry 28A80 (Primary) 39B62 (Secondary) |
| url | https://arxiv.org/abs/2603.03174 |