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arXiv · 1811.03872

Embedding Properties of sets with finite box-counting dimension

Alexandros Margaris·James C. Robinson
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Abstract

In this paper we study the regularity of embeddings of finite--dimensional subsets of Banach spaces into Euclidean spaces. In 1999, Hunt and Kaloshin [Nonlinearity 12 1263-1275] introduced the thickness exponent and proved an embedding theorem for subsets of Hilbert spaces with finite box--counting dimension. In 2009, Robinson [Nonlinearity 22 711-728] defined the dual thickness and extended the result to subsets of Banach spaces. Here we prove a similar result for subsets of Banach spaces, using the thickness rather than the dual thickness. We also study the relation between the box-counting dimension and these two thickness exponents for some particular subsets of $\ell_{p}$.

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Alexandros Margaris, James C. Robinson. 2019-04-10. Embedding Properties of sets with finite box-counting dimension. https://doi.org/10.1088/1361-6544%2Fab1b7f

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