The Assouad dimension of the Sierpiński triangle is equal to its Hausdorff dimension, ${\displaystyle \alpha ={\frac {\log(3)}{\log(2)))}$. In the illustration, we see that for a particular choice of r, R, and x,
${\displaystyle N_{r}(B_{R}(x)\cap E)=3=2^{\alpha }=\left({\frac {R}{r))\right)^{\alpha }.}$
For other choices, the constant C may be greater than 1, but is still bounded.

In mathematics — specifically, in fractal geometry — the Assouad dimension is a definition of fractal dimension for subsets of a metric space. It was introduced by Patrice Assouad in his 1977 PhD thesis and later published in 1979,[1] although the same notion had been studied in 1928 by Georges Bouligand.[2] As well as being used to study fractals, the Assouad dimension has also been used to study quasiconformal mappings and embeddability problems.

Definition

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 ${\displaystyle M\geq 1}$.[3]

Let ${\displaystyle (X,d)}$ be a metric space, and let E be a non-empty subset of X. For 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 cover the set E. The Assouad dimension of E is defined to be the infimal ${\displaystyle \alpha \geq 0}$ for which there exist positive constants C and ${\displaystyle \rho }$ so that, whenever

${\displaystyle 0
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.

References

1. ^ 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 (in French). 288 (15): A731–A734. ISSN 0151-0509. MR532401
2. ^ Bouligand, Georges (1928). "Ensembles impropres et nombre dimensionnel". Bulletin des Sciences Mathématiques (in French). 52: 320–344.
3. ^ Robinson, James C. (2010). Dimensions, Embeddings, and Attractors. Cambridge University Press. p. 85. ISBN 9781139495189.
4. ^ Le Donne, Enrico; Rajala, Tapio (2015). "Assouad dimension, Nagata dimension, and uniformly close metric tangents". Indiana University Mathematics Journal. 64 (1): 21–54. arXiv:1306.5859. doi:10.1512/iumj.2015.64.5469.
5. ^ a b Luukkainen, Jouni (1998). "Assouad dimension: antifractal metrization, porous sets, and homogeneous measures". Journal of the Korean Mathematical Society. 35 (1): 23–76. ISSN 0304-9914.