Intermediate dimensions -- a survey

TL;DR

This survey reviews the recently introduced $ heta$-intermediate dimensions, a continuum of fractal dimensions bridging Hausdorff and box-counting dimensions, highlighting their properties and examples.

Contribution

It consolidates diverse properties of $ heta$-intermediate dimensions and illustrates their behavior through examples, providing a comprehensive overview.

Findings

$ heta$-intermediate dimensions interpolate between Hausdorff and box-counting dimensions.

They exhibit diverse properties that differ from classical dimensions.

Examples demonstrate the range and behavior of these intermediate dimensions.

Abstract

This article surveys the $\theta$-intermediate dimensions that were introduced recently which provide a parameterised continuum of dimensions that run from Hausdorff dimension when $\theta=0$ to box-counting dimensions when $\theta=1$. We bring together diverse properties of intermediate dimensions which we illustrate by examples.