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# <title>arXiv Query: search_query=&id_list=2206.06921&start=0&max_results=10</title>
# <updated>2025-11-11T18:30:58Z</updated>
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# <id>http://arxiv.org/abs/2206.06921v3</id>
# <title>Attainable forms of Assouad spectra</title>
# <updated>2025-01-30T01:23:42Z</updated>
# <link href="https://arxiv.org/abs/2206.06921v3"/>
# <link href="https://arxiv.org/pdf/2206.06921v3"/>
# <summary>Let $d\in\mathbb{N}$ and let $\varphi\colon(0,1)\to[0,d]$. We prove that there exists a set $F\subset\mathbb{R}^d$ such that $\operatorname{dim}_A^θF=\varphi(θ)$ for all $θ\in(0,1)$ if and only if for every $0<λ<θ<1$, \[0\leq (1-λ)\varphi(λ)-(1-θ)\varphi(θ)\leq (θ-λ)\varphi\Bigl(\fracλθ\Bigr).\] 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.</summary>
# <category term="math.CA" scheme="http://arxiv.org/schemas/atom"/>
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# <published>2022-06-14T15:35:46Z</published>
# <arxiv:comment>25 pages, 5 figures. v3: Many typo fixes and substantially improved exposition, especially in non-monotonic construction. Some numbering changes from v2. Results unchanged</arxiv:comment>
# <arxiv:primary_category term="math.CA"/>
# <arxiv:journal_ref>Indiana Univ. Math. J. 73 (2024), 1331-1356</arxiv:journal_ref>
# <author>
# <name>Alex Rutar</name>
# </author>
# <arxiv:doi>10.1512/iumj.2024.73.9928</arxiv:doi>
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