In mathematics, the Assouad–Nagata dimension (sometimes simply Nagata dimension) is a notion of dimension for metric spaces,[1][2] introduced by Jun-iti Nagata in 1958[3] and reformulated by Patrice Assouad in 1982, who introduced the now-usual definition.[4]
Definition
editThe 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
editCompare 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.[5]
References
edit- ^ Cobzaş, Ş.; Miculescu, R.; Nicolae, A. (2019). Lipschitz functions. Cham, Switzerland: Springer. p. 308. ISBN 978-3-030-16488-1.
- ^ Lang, Urs; Schlichenmaier, Thilo (2005). "Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions". International Mathematics Research Notices. 2005 (58): 3625. arXiv:math/0410048. doi:10.1155/IMRN.2005.3625. S2CID 119683379.
- ^ Nagata, J. (1958). "Note on dimension theory for metric spaces". Fundamenta Mathematicae. 45: 143–181. doi:10.4064/fm-45-1-143-181.
- ^ Assouad, P. (January 4, 1982). "Sur la distance de Nagata". Comptes Rendus de l'Académie des Sciences, Série I (in French). 294 (1): 31–34.
- ^ Le Donne, Enrico; Rajala, Tapio (2015). "Assouad dimension, Nagata dimension, and uniformly close metric tangents". Indiana University Mathematics Journal. 64 (1): 21–54. arXiv:1306.5859. doi:10.1512/iumj.2015.64.5469. S2CID 55039643.
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