Random Cutouts of the Unit Cube with I.U.D Centers

Abstract

Consider the random open balls \(B_n(\omega):=B(\omega_n,r_n)\) with their centers \(\omega_n\) being i.u.d. on the \(d\)-dimensional unit cube \([0,1]^d\) and with their radii \(r_n\sim cn^{-\frac{1}{d}}\) for some constant \(0<c<(\beta(d))^{-\frac{1}{d}}\), where \(\beta(d)\) is the volume of the \(d\) dimensional unit ball. We call \([0,1]^d-\bigcup_{n=1}^{\infty} B_n(\omega)\) a random cutout set. In this paper, we present an exposition of the Zähle cutout model by a detailed study of such a random cutout set for the purpose of teaching and learning. We show that with probability one the Hausdorff dimension of such random cut-out set is at most \(d(1-\beta(d)c^d)\) and frequently equals \(d(1-\beta(d)c^d)\).