Assouad–Nagata dimension

In mathematics, the Assouad–Nagata dimension (sometimes simply 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 $(X, d)$ is defined as the smallest integer $n$ for which there exists a constant $C > 0$ such that for all $r > 0$ the space $X$ has a $Cr$-bounded covering with $r$-multiplicity at most $n + 1$. Here $Cr$-bounded means that the diameter of each set of the covering is bounded by $Cr$, and $r$-multiplicity is the infimum of integers $k ≥ 0$ such that each subset of $X$ with diameter at most $r$ has a non-empty intersection with at most $k$ 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 $(X, d)$ is the smallest integer $n$ for which there exists a constant $c > 0$ such that for every $r > 0$, the covering of $X$ by $r$-balls has a refinement with $cr$-multiplicity at most $n + 1$.

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 $n$ if it is at most $n$-dimensional at microscopic scales, and asymptotic dimension at most $n$ if it looks at most $n$-dimensional upon zooming out as far as you need. To have Assouad–Nagata dimension at most $n$, a space has to look at most $n$-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.