Embedding properties of sets with finite box-counting dimension
Alexandros Margaris, James C. Robinson · Nonlinearity · 2019
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 (1999 Nonlinearity 12 1263–75) introduced the thickness exponent and proved an embedding theorem for subsets of Hilbert spaces with finite box-counting dimension. In 2009, Robinson (2009 Nonlinearity 22 711–28) 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 .