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. It was defined earlier by Georges Bouligand (1928). As well as being used to study fractals, the Assouad dimension has also been used to study quasiconformal mappings and embeddability problems.
Definition.
Let formula_1 be a metric space, and let formula_2 be a non-empty subset of formula_3. For formula_4, let formula_5 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 formula_2. The Assouad dimension of formula_2 is defined to be the infimal formula_8 for which there exist positive constants formula_9 and formula_10 so that, whenever
the following bound holds:
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 ()"n".