Assouad dimension

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. As well as being used to study fractals, the Assouad dimension has also been used to study quasiconformal mappings and embeddability problems.

Definition

Let (X, d) be a metric space, and let E be a non-empty subset of X. For r > 0, let Nr(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 E. The Assouad dimension of E is defined to be the infimal α ≥ 0 for which there exist positive constants C and ρ so that, whenever

0 < r < R \leq \rho,

the following bound holds:

\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.

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