Sets with arbitrary Hausdorff and packing scales in infinite dimensional Banach spaces

TL;DR

This paper constructs compact subsets with prescribed Hausdorff and packing measures in infinite-dimensional Banach spaces, answering a question about the existence of metric spaces with arbitrary measure scales.

Contribution

It demonstrates the existence of compact sets with arbitrary Hausdorff and packing measures in Banach spaces, extending previous results to infinite dimensions.

Findings

Existence of compact sets with finite positive Hausdorff and packing measures for given functions

Embedding of these sets into any infinite-dimensional Banach space

Positive answer to Fan's question on metric spaces with arbitrary scales

Abstract

For every couple of Hausdorff functions $ \psi$ and $\varphi $ verifying some mild assumptions, there exists a compact subset $ K $ of the Baire space such that the $ \varphi$-Hausdorff measure and the $ \psi$-packing measure on $ K$ are both finite and positive. Such examples are then embedded in any infinite dimensional Banach space to answer positively a question of Fan on the existence of metric spaces with arbitrary scales.