The set of real numbers left uncovered by random covering intervals

Summary Random covering intervals are placed on the real line in a Poisson manner. Lebesgue measure governs their (random) locations and an arbitrary measure μ governs their (random) lengths. The uncovered set is a regenerative set in the sense of Hoffmann-J?rgensen's generalization of regenerative phenomena introduced by Kingman. Thus, as has previously been obtained by Mandelbrot, it is the closure of the image of a subordinator —one that is identified explicitly. Well-known facts about subordinators give Shepp's necessary and sufficient condition on μ for complete coverage and, when the coverage is not complete, a formula for the Hausdorff dimension of the uncovered set. The method does not seem to be applicable when the covering is not done in a Poisson manner or if the line is replaced by the plane or higher dimensional space. This work does not represent a collaboration among the three authors, but rather is an outgrowth of a discovery that the first author on one hand and the second and third authors on the other hand had proved identical results via similar methods. Based on part of the first author's PhD dissertation written under Kenneth J. Hochberg at Case Western Reserve University. Research partially supported by National Science Foundation Grant MCS 78-01168.