mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Assouad–Nagata dimension (sometimes simply Nagata dimension) is a notion of

dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...

for

metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...

s, introduced by

Jun-iti Nagata was a Japanese mathematician specializing in topology. In 1956, Jun-iti Nagata earned his PhD from Osaka University under the direction of Kiiti Morita. He was the author of two standard graduate texts in topology: ''Modern Dimension Theory'' ...

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 In metric geometry, asymptotic dimension of a metric space is a large-scale analog of Lebesgue covering dimension. The notion of asymptotic dimension was introduced by Mikhail Gromov in his 1993 monograph ''Asymptotic invariants of infinite groups ...

. 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 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, al ...

References

Metric geometry Dimension theory {{Mathanalysis-stub