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Turing degrees of reals of positive effective packing dimension. (English) Zbl 1191.68304
Summary: A relatively longstanding question in algorithmic randomness is Jan Reimann’s question whether there is a Turing cone of broken dimension. That is, is there a real \(A\) such that \(\{B: B \leqslant _T A\}\) contains no 1-random real, yet contains elements of nonzero effective Hausdorff dimension? We show that the answer is affirmative if Hausdorff dimension is replaced by its inner analogue packing dimension. We construct a minimal degree of effective packing dimension 1. This leads us to examine the Turing degrees of reals with positive effective packing dimension. Unlike effective Hausdorff dimension, this is a notion of complexity which is shared by both random and sufficiently generic reals. We provide a characterization of the c.e. array noncomputable degrees in terms of effective packing dimension.

MSC:
68Q05 Models of computation (Turing machines, etc.) (MSC2010)
03D28 Other Turing degree structures
03D32 Algorithmic randomness and dimension
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[1] Athreya, K.; Hitchcock, J.; Lutz, J.; Mayordomo, E., Effective strong dimension in algorithmic information and computational complexity, SIAM journal on computing, 37, 3, 671-705, (2007) · Zbl 1144.68029
[2] Barzdins, J., Complexity of programs to determine whether natural numbers not greater than n belong to a recursively enumerable set, Soviet mathematics doklady, 9, 1251-1254, (1968) · Zbl 0193.31601
[3] Bienvenu, L.; Doty, D.; Stephan, F., Constructive dimension and weak truth-table degrees. conference of computability in Europe 2007, ()
[4] C. Conidis, PhD Thesis, University of Chicago, in preparation
[5] de Leeuw, K.; Moore, E.F.; Shannon, C.E.; Shapiro, N., Computability by probabilistic machines, (), 183-212
[6] Doty, D., Dimension extractors and optimal decompression, () · Zbl 1166.68019
[7] R. Downey, D. Hirschfeldt, Algorithmic Randomness and Complexity, Springer-Verlag, in press · Zbl 1221.68005
[8] Downey, R.; Hirschfeldt, D.; Nies, A.; Terwijn, S., Calibrating randomness, Bulletin of symbolic logic, 3, 411-491, (2006) · Zbl 1113.03037
[9] Downey, R.; Jockusch, C.; Stob, M., Array nonrecursive sets and multiple permitting arguments, (), 141-174 · Zbl 0713.03020
[10] Downey, R.; Jockusch, C.; Stob, M., Array nonrecursive degrees and genericity, (), 93-105 · Zbl 0849.03029
[11] Falconer, K., Fractal geometry, mathematical foundations & applications, (1992), Wiley & Sons
[12] Y. Gabbay, Double jump inversions and strong minimal covers in the Turing degrees, PhD Thesis, Cornell University, 2004
[13] Hausdorff, F., Dimension und äußeres maß, Mathematische annalen, 79, 157-179, (1919) · JFM 46.0292.01
[14] Ishmukhametov, S., Weak recursive degrees and a problem of spector, (), 81-88 · Zbl 0951.03036
[15] Kummer, M., Kolmogorov complexity and instance complexity of recursively enumerable sets, SIAM journal of computing, 25, 1123-1143, (1996) · Zbl 0859.03015
[16] Lerman, M., Degrees of unsolvability. local and global theory, (), (xiii+307 pp.) · Zbl 0542.03023
[17] Li, M.; Vitányi, P., Kolmogorov complexity and its applications, (1993), Springer-Verlag
[18] Lutz, J., Category and measure in complexity classes, SIAM journal on computing, 19, 1100-1131, (1990) · Zbl 0711.68046
[19] Lutz, J., The dimensions of individual strings and sequences, Information and computation, 187, 49-79, (2003) · Zbl 1090.68053
[20] Mayordomo, E., A Kolmogorov complexity characterization of constructive Hausdorff dimension, Information processing letters, 84, 1-3, (2002) · Zbl 1045.68570
[21] Miller, J.S.; Nies, A., Randomness and computability: open questions, Bulletin symbolic logic, 12, 3, 390-410, (2006) · Zbl 1169.03033
[22] A. Nies, Computability and Randomness, Oxford University Press, in preparation
[23] A. Nies, J. Reimann, A lower cone in the wtt degrees of non-integral effective dimension, in: Proceedings of IMS Workshop on Computational Prospects of Infinity, Singapore, 2006 · Zbl 1158.03026
[24] J. Reimann, Computability and fractal dimension, Ph.D. thesis, Universität Heidelberg, 2004 · Zbl 1080.03031
[25] Sacks, G.E., Forcing with perfect closed sets, (), 331-355
[26] Soare, R., Recursively enumerable sets and degrees, (1987), Springer Berlin · Zbl 0623.03042
[27] Staiger, L., Kolmogorov complexity and Hausdorff dimension, Information and computation, 103, 159-194, (1993) · Zbl 0789.68076
[28] Terwijn, S.; Zambella, D., Algorithmic randomness and lowness, Journal of symbolic logic, 66, 1199-1205, (2001) · Zbl 0990.03033
[29] D. Zambella, On sequences with simple initial segments, ILLC technical report, ML-1990-05, University of Amsterdam, 1990
[30] M. Zimand, Two sources are better than one for increasing the Kolmogorov complexity of infinite sequences, in: Proceedings CSR 2008, Moscow, June 2008 · Zbl 1143.68020
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