On the intermediate dimensions of concentric spheres and related sets

TL;DR

This paper computes the intermediate dimensions, which interpolate between Hausdorff and box dimensions, for specific sets like concentric spheres, isolated points, and attenuated sine curves, providing explicit examples in Euclidean spaces.

Contribution

It offers the first explicit calculations of intermediate dimensions for sets of concentric spheres and related structures, expanding understanding of these dimensions.

Findings

Intermediate dimensions of concentric spheres are explicitly calculated.

Results include intermediate dimensions of isolated points on spheres.

Analysis of attenuated topologist's sine curves' dimensions.

Abstract

The intermediate dimensions are a family of dimensions introduced in 2019 by Falconer, Fraser, and Kempton [arXiv:1811.06493] to interpolate between the Hausdorff dimension and the box dimension. To date, there are limited examples of explicit calculations of the intermediate dimensions of interesting sets. We calculate the intermediate dimensions of sets of concentric spheres converging to the origin in Euclidean spaces. We also consider related sets including isolated points on concentric spheres and attenuated topologist's sine curves.