论文标题
可达到的Assouad Spectra形式
Attainable forms of Assouad spectra
论文作者
论文摘要
令$ d \ in \ mathbb {n} $，让$φ\ colon（0,1）\ to [0，d] $。我们证明存在一个集合$ f \ subset \ mathbb {r}^d $，以便所有$θ\ in（0,1）$ in（0,1）$，并且仅在每一个$ 0 <λ<λ<θ<θ<θ<1 $，\ [0 \θ<1 $，\ [0 \ [0 \ fe [0,1）$ in（0,1）$中$ \ operatoTorname {dim} _a _a^θf=φ（θ）$ [0,1）$ （1-λ）φ（λ） - （1-θ）φ（θ）\ leq（θ-λ）φ\ bigl（\fracλθ\ bigr）。
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.
