Attainable forms of intermediate dimensions

TL;DR

This paper characterizes exactly when a function can represent the intermediate dimensions of a set in Euclidean space, using conditions on the function's Dini derivatives, and demonstrates sharpness with a specific set construction.

Contribution

It provides a necessary and sufficient condition for a function to be the intermediate dimension profile of a set, advancing understanding of fractal dimensions.

Findings

Established a sharp criterion based on Dini derivatives for intermediate dimensions.

Constructed a homogeneous Moran set to demonstrate the sharpness of the condition.

Bridged the gap between theoretical conditions and realizability of dimension functions.

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.