Cubes, side lengths and centres

TL;DR

This paper explores the geometric and measure-theoretic properties of fractal cubes, spheres, and centers in high-dimensional spaces, extending classical results to fractal and exceptional sets, and discusses connections to Furstenberg-type problems.

Contribution

It introduces a fractal analogue of classical sphere and cube covering problems, analyzing Hausdorff dimension, measure, and exceptional sets, and relates these to Furstenberg-type conjectures.

Findings

Fractal cubes and spheres can have full Hausdorff dimension.

Certain fractal sets of centers and radii have positive Lebesgue measure.

Discussion of Furstenberg-type examples in the context of fractal cube and circle sets.

Abstract

It is known that in $\mathbb{R}^n,n\geq 2$, a compact set which contains $n-1$ spheres with all radii in $[1/2,1]$ or with all possible centres in $[0,1]^n$ has full Hausdorff dimension. In fact the later set has positive Lebesgue measure. In this paper we consider a similar problem with sphere replacing by fractal cubes. The radii set and the centre set are also considered to be fractal sets. In addition we discuss the exceptional set in the setting of general largeness. In the end, an Furstenberg type example is discussed which can be somehow considered as the Furstenberg $\times 2$, $\times 3$ set conjecture (now theorem) in the setting of cubes/circles sets considered here.