View source for Assouad–Nagata dimension

In [[Mathematics|mathematics]], the '''Assouad–Nagata dimension''' (sometimes simply '''Nagata dimension''') is a notion of [[Physics:Dimension|dimension]] for [[Metric space|metric space]]s,<ref>{{cite book |last1=Cobzaş |first1=Ş. |last2=Miculescu |first2=R. |last3=Nicolae |first3=A. |title=Lipschitz functions |date=2019 |publisher=Springer |location=Cham, Switzerland |isbn=978-3-030-16488-1 |page=308}}</ref><ref>{{cite journal|last1=Lang|first1=Urs|last2=Schlichenmaier|first2=Thilo|title=Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions|journal=[[International Mathematics Research Notices]] |year=2005 |volume=2005 |issue=58 |page=3625 |arxiv=math/0410048|doi=10.1155/IMRN.2005.3625|doi-access=<!-- Not free-->|s2cid=119683379 }}</ref> introduced by [[Biography:Jun-iti Nagata|Jun-iti Nagata]] in 1958<ref>{{cite journal |last1=Nagata |first1=J. |title=Note on dimension theory for metric spaces |journal=[[Fundamenta Mathematicae]] |date=1958 |volume=45 |pages=143–181 |doi=10.4064/fm-45-1-143-181 |doi-access=free}}</ref> and reformulated by Patrice Assouad in 1982, who introduced the now-usual definition.<ref>{{cite journal |last1=Assouad |first1=P. |title=Sur la distance de Nagata |journal=Comptes Rendus de l'Académie des Sciences, Série I |date=January 4, 1982 |volume=294 |issue=1 |pages=31–34 |url=https://gallica.bnf.fr/ark:/12148/bpt6k5533029f/f49.item |lang=fr}}</ref>
==Definition==

The '''Assouad–Nagata dimension''' of a metric space {{math|(''X'', ''d'')}} is defined as the smallest integer {{mvar|n}} for which there exists a constant {{math|''C'' > 0}} such that for all {{math|''r'' > 0}} the space {{mvar|X}} has a {{math|''Cr''}}-bounded [[Cover (topology)|covering]] with {{mvar|r}}-multiplicity at most {{math|''n'' + 1}}. Here ''{{mvar|Cr}}-bounded'' means that the diameter of each set of the covering is bounded by {{mvar|Cr}}, and ''{{mvar|r}}-multiplicity'' is the infimum of integers {{math|''k'' ≥ 0}} such that each subset of {{mvar|X}} with diameter at most {{mvar|r}} has a non-empty intersection with at most {{mvar|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 {{math|(''X'', ''d'')}} is the smallest integer {{mvar|n}} for which there exists a constant {{math|''c'' > 0}} such that for every {{math|''r'' > 0}}, the covering of {{mvar|X}} by {{mvar|r}}-balls has a refinement with {{mvar|cr}}-multiplicity at most {{math|''n'' + 1}}.

==Relationship to other notions of dimension==

Compare the similar definitions of Lebesgue covering dimension and [[Asymptotic dimension|asymptotic dimension]].  A space has Lebesgue covering dimension at most {{mvar|n}} if it is at most {{mvar|n}}-dimensional at microscopic scales, and asymptotic dimension at most {{mvar|n}} if it looks at most {{mvar|n}}-dimensional upon zooming out as far as you need.  To have Assouad–Nagata dimension at most {{mvar|n}}, a space has to look at most {{mvar|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]].<ref>{{cite journal |last1=Le Donne |first1=Enrico |last2=Rajala |first2=Tapio |title=Assouad dimension, Nagata dimension, and uniformly close metric tangents |journal=[[Organization:Indiana University Mathematics Journal|Indiana University Mathematics Journal]] |date=2015 |volume=64 |issue=1 |pages=21–54 |doi=10.1512/iumj.2015.64.5469 |arxiv=1306.5859|s2cid=55039643 }}</ref>

==References==
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[[Category:Metric geometry]]
[[Category:Dimension theory]]


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