Archive for Mathematical Logic 48 (3-4):257-263 (2009)

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
Keywords Random closed sets  Random recursion
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DOI 10.1007/s00153-009-0126-6
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