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Martin-Löf randomness and Galton-Watson processes. (English) Zbl 1247.03085
Summary: The members of Martin-Löf random closed sets under a distribution studied by Barmpalias et al. are exactly the infinite paths through Martin-Löf random Galton-Watson trees with survival parameter $$\frac 23$$. To be such a member, a sufficient condition is to have effective Hausdorff dimension strictly greater than $$\gamma = \log_2 \frac 32$$, and a necessary condition is to have effective Hausdorff dimension greater than or equal to $$\gamma$$.

##### MSC:
 03D32 Algorithmic randomness and dimension 68Q30 Algorithmic information theory (Kolmogorov complexity, etc.) 60C05 Combinatorial probability
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##### References:
 [1] Logan Axon, Random closed sets and probability, Doctoral Dissertation, University of Notre Dame, 2010. · Zbl 06835068 [2] Barmpalias, George; Brodhead, Paul; Cenzer, Douglas; Dashti, Seyyed; Weber, Rebecca, Algorithmic randomness of closed sets, J. logic comput., 17, 6, 1041-1062, (2007), MR 2376074 (2008m:68069) · Zbl 1155.03031 [3] Diamondstone, David; Kjos-Hanssen, Bjørn, Members of random closed sets, (), 144-153 · Zbl 1233.03049 [4] Falconer, Kenneth, Fractal geometry, Mathematical foundations and applications, (1990), John Wiley & Sons Ltd. Chichester, MR 1102677 (92j:28008) · Zbl 0689.28003 [5] Gács, Péter, On the relation between descriptional complexity and algorithmic probability, Theoret. comput. sci., 22, 1-2, 71-93, (1983), MR 693050 (84h:60010) · Zbl 0562.68035 [6] Greenberg, Noam; Miller, Joseph S., Lowness for Kurtz randomness, J. symbolic logic, 74, 2, 665-678, (2009), MR 2518817 (2010b:03050) · Zbl 1168.03033 [7] Hawkes, John, Trees generated by a simple branching process, J. London math. soc. (2), 24, 2, 373-384, (1981), MR 631950 (83b:60072) · Zbl 0468.60081 [8] Kjos-Hanssen, Bjørn, Infinite subsets of random sets of integers, Math. res. lett., 16, 1, 103-110, (2009), MR 2480564 (2010b:03051) · Zbl 1179.03061 [9] Kjos-Hanssen, Bjørn; Merkle, Wolfgang; Stephan, Frank, Kolmogorov complexity and the recursion theorem, Trans. amer. math. soc., 363, 10, 5465-5480, (2011) · Zbl 1236.03032 [10] Kjos-Hanssen, Bjørn; Nerode, Anil, Effective dimension of points visited by Brownian motion, Theoret. comput. sci., 410, 4-5, 347-354, (2009), MR 2493984 (2009k:68100) · Zbl 1158.68017 [11] Lutz, Jack H., (), 902-913, MR 1795945 (2001g:68046) [12] Lyons, Russell, Random walks and percolation on trees, Ann. prob., 18, 3, 931-958, (1990), MR 1062053 (91i:60179) · Zbl 0714.60089 [13] Mattila, Pertti, Geometry of sets and measures in Euclidean spaces, (), MR 1333890 (96h:28006) · Zbl 0911.28005 [14] Peter Mörters, Yuval Peres, Brownian Motion, Draft version of May 25, 2008. http://www.stat.berkeley.edu/ peres/. [15] Jan Reimann, Computability and Fractal Dimension, Doctoral Dissertation, Universität Heidelberg, 2004. · Zbl 1080.03031 [16] Reimann, Jan, Effectively closed sets of measures and randomness, Ann. pure appl. logic, 156, 1, 170-182, (2008), MR 2474448 (2010a:03043) · Zbl 1153.03021 [17] Reimann, Jan; Stephan, Frank, Effective Hausdorff dimension, (), 369-385, MR 2143904 (2006b:03052) · Zbl 1098.03050 [18] Uspensky, V.A.; Shen, A., Relations between varieties of Kolmogorov complexities, Math. systems theory, 29, 3, 271-292, (1996), MR 1374498 (97c:68074) · Zbl 0849.68059 [19] Zvonkin, A.K.; Levin, L.A., The complexity of finite objects and the basing of the concepts of information and randomness on the theory of algorithms, Uspekhi mat. nauk, 25, (1970), no. 6(156), 85-127 (in Russian) MR 0307889 (46 #7004) · Zbl 0222.02027
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