Cut-out sets and the Zipf law for fractal voids

“Cut-out sets” are fractals that can be obtained by removing a sequence of disjoint regions from an initial region of dd-dimensional euclidean space. Conversely, a description of some fractals in terms of their void complementary set is possible. The essential property of a sequence of fractal voids is that their sizes decrease as a power law, that is, they follow Zipf’s law. We prove the relation between the box dimension of the fractal set (for d≤3d≤3) and the exponent of the Zipf law for convex   voids; namely, if the Zipf law exponent ee is such that 1