Assouad dimension

The Assouad dimension of ${\displaystyle X,d_{A}(X)}$, is the infimum of all ${\displaystyle s}$ such that ${\displaystyle (X,\varsigma )}$ is ${\displaystyle (M,s)}$-homogeneous for some ${\displaystyle M\geq 1}$.[1]

Let ${\displaystyle (X,d)}$ be a metric space, and let ${\displaystyle E}$ be a non-empty subset of ${\displaystyle X}$. For ${\displaystyle r>0}$, let ${\displaystyle N_{r}(E)}$ denote the least number of metric open balls of radius less than or equal to r with which it is possible to open cover the set ${\displaystyle E}$. The Assouad dimension of ${\displaystyle E}$ is defined to be the infimal ${\displaystyle \alpha \geq 0}$ for which there exist positive constants ${\displaystyle C}$ and ${\displaystyle \rho }$ so that, whenever

${\displaystyle 0<r<R\leq \rho ,}$

the following bound holds:

${\displaystyle \sup _{x\in E}N_{r}(B_{R}(x)\cap E)\leq C\left({\frac {R}{r}}\right)^{\alpha }.}$

The intuition underlying this definition is that, for a set E with "ordinary" integer dimension n, the number of small balls of radius r needed to cover the intersection of a larger ball of radius R with E will scale like (R/r)ⁿ.

• Assouad, Patrice (1979). "Étude d'une dimension métrique liée à la possibilité de plongements dans Rⁿ". Comptes Rendus de l'Académie des Sciences, Série A-B. 288 (15): A731–A734. ISSN 0151-0509. MR532401
• Bouligand, M.G. (1928). "Ensembles impropres et nombredimensionnel", Bulletin des Sciences Mathématiques 52, pp.320–344.