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.
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R. Daniel Mauldin and A. P. McLinden were supported in part by NSF grants DMS 0700831 and DMS 0652450.
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Daniel Mauldin, R., McLinden, A.P. Random closed sets viewed as random recursions. Arch. Math. Logic 48, 257–263 (2009). https://doi.org/10.1007/s00153-009-0126-6
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DOI: https://doi.org/10.1007/s00153-009-0126-6