On concentric fractal spheres and 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 through consecutive disjoint similar copies 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-Hölder map between shells and constitutes an addition to a general programme of research proposed by Fraser (2021 Nonlinearity 34 3251–70).