# Assouad dimension

## From Wikipedia, the free encyclopedia

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

The

Assouad dimensionof , is the infimum of all such that ς is -homogeneous for some .^{[1]}

Let (*X*, *d*) be a metric space, and let *E* be a non-empty subset of *X*. For *r* > 0, let *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 *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

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 (*R*/*r*)^{n}.

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