Hausdorff dimension of the random middle third Cantor set

Abstract

The iterative elimination of the middle spacing in the random division of intervals with two points “at random” — in the narrow sense of uniformly distributed — generates a random middle Cantor set. We compute the Hausdorff dimension (which intuitively evaluates how “dense” a set is) of the random middle third Cantor set, and we verify that although the deterministic middle third Cantor set is the expectation of the random middle third Cantor set, it is more dense than its stochastic counterpart. This can be explained by the dependence of order statistics