Assouad–Nagata dimension
In mathematics, the Assouad–Nagata dimension is a notion of dimension for metric spaces, introduced by Jun-iti Nagata in 1958 and reformulated by Patrice Assouad in 1982, who introduced the now-usual definition.
Definition
The Assouad–Nagata dimension of a metric space is defined as the smallest integer for which there exists a constant such that for all the space has a -bounded covering with -multiplicity at most. Here -bounded means that the diameter of each set of the covering is bounded by, and -multiplicity is the infimum of integers such that each subset of with diameter at most has a non-empty intersection with at most members of the covering.This definition can be rephrased to make it more similar to that of the Lebesgue covering dimension. The Assouad–Nagata dimension of a metric space is the smallest integer for which there exists a constant such that for every, the covering of by -balls has a refinement with -multiplicity at most.
Relationship to other notions of dimension
Compare the similar definitions of Lebesgue covering dimension and asymptotic dimension. A space has Lebesgue covering dimension at most if it is at most -dimensional at microscopic scales, and asymptotic dimension at most if it looks at most -dimensional upon zooming out as far as you need. To have Assouad–Nagata dimension at most, a space has to look at most -dimensional at every possible scale, in a uniform way across scales.The Nagata dimension of a metric space is always less than or equal to its Assouad dimension.