Random closed sets viewed as random recursions

Abstract

It is known that the box dimension of any Martin-Löf random closed set of ${\{0,1\}^\mathbb{N}}$ is ${\log_2(\frac{4}{3})}$. Barmpalias et al. [J Logic Comput 17(6):1041–1062, 2007] gave one method of producing such random closed sets and then computed the box dimension, and posed several questions regarding other methods of construction. We outline a method using random recursive constructions for computing the Hausdorff dimension of almost every random closed set of ${\{0,1\}^\mathbb{N}}$, and propose a general method for random closed sets in other spaces. We further find both the appropriate dimensional Hausdorff measure and the exact Hausdorff dimension for such random closed sets.