Attainable forms of intermediate dimensions

Abstract

The intermediate dimensions are a family of dimensions which interpolate between the Hausdorff and box dimensions of sets. We prove a necessary and sufficient condition for a given function $h(\theta )$ to be realized as the intermediate dimensions of a bounded subset of ${\mathbb{R}}^d$. This condition is a straightforward constraint on the Dini derivatives of $h(\theta )$, which we prove is sharp using a homogeneous Moran set construction.