On concentric fractal spheres and spiral shells

TL;DR

This paper studies the fractal and dimension properties of concentric spheres and applies these findings to classify spiral shells in higher dimensions, advancing understanding in geometric analysis.

Contribution

It introduces new dimension estimates for fractal spheres and uses them to classify spiral shells via quasiconformal maps, extending existing mathematical frameworks.

Findings

Calculated box dimension and Assouad spectrum of fractal spheres.

Proved fractal spheres cannot be shrunk to a point at polynomial rates.

Established bi-Hölder equivalences between spiral shells.

Abstract

We investigate dimension-theoretic properties of concentric topological spheres, which are fractal sets emerging both in pure and applied mathematics. We calculate the box dimension and Assouad spectrum of such collections, and use them to prove that fractal spheres cannot be shrunk into a point at a polynomial rate. We also apply these dimension estimates to quasiconformally classify certain spiral shells, a generalization of planar spirals in higher dimensions. This classification also provides a bi-H\"older map between shells, and constitutes an addition to a general programme of research proposed by J. Fraser.