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.
The Assouad dimension of , is the infimum of all such that is -homogeneous for some .[3]
Let be a metric space, and let E be a non-empty subset of X. For r > 0, let 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 for which there exist positive constants C and so that, whenever
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.