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 d -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 ≤ 3 ) and the exponent of the Zipf law for convex voids; namely, if the Zipf law exponent e is such that 1 < e < d / ( d − 1 ) and, in addition, we forbid the appearance of degenerate void shapes, we prove that the corresponding cut-out set has box dimension d / e (such that d − 1 < d / e < d ). We explore various physical applications of this result, in particular, the application to the description of the cosmic structure using “cosmic foam” models.