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

Abstract

For every couple of Hausdorff functions $\psi$ and $\phi$ verifying some mild assumptions, there exists a compact subset $K$ of the Baire space such that the $\phi$-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.