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A real of strictly positive effective packing dimension that does not compute a real of effective packing dimension one

  • Chris J. Conidis (a1)

Abstract

Recently, the Dimension Problem for effective Hausdorff dimension was solved by J. Miller in [14], where the author constructs a Turing degree of non-integral Hausdorff dimension. In this article we settle the Dimension Problem for effective packing dimension by constructing a real of strictly positive effective packing dimension that does not compute a real of effective packing dimension one (on the other hand, it is known via [10. 3. 7] that every real of strictly positive effective Hausdorff dimension computes reals whose effective packing dimensions are arbitrarily close to, but not necessarily equal to, one).

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[1] Athreya, K. B., Hitchcock, J. M., Lutz, J. H., and Mayordomo, E., Effective strong dimension in algorithmic information and computational complexity, SIAM Journal on Computing, vol. 37 (2007), pp. 671705.
[2] Barak, B., Impagliazzo, R., and Wigderson, A., Extracting randomness using few independent sources, SIAM Journal on Computing, vol. 36 (2006), pp. 10951118.
[3] Bienvenu, L., Doty, D., and Stephan, F., Constructive dimension and Turing degrees, Theory of Computing Systems, vol. 45 (2009), pp. 740755.
[4] Bourgain, J., Katz, N., and Tao, T., A finite sum-product estimate in finite fields, and applications, Geometric and Functional Analysis, vol. 14 (2004), pp. 2757.
[5] Conidis, C. J., Effective packing dimension of -classes, Proceedings of the American Mathematical Society, vol. 136 (2008), pp. 36553662.
[6] Downey, R. G. and Greenberg, N., Turing degrees of reals of positive packing dimension, Information Processing Letters, vol. 108 (2008), pp. 198203.
[7] Downey, R. G. and Hirshfeldt, D. R., Algorithmic randomness and complexity, Springer-Verlag, 2010.
[8] Downey, R. G. and Ng, K. M., Effective packing dimension and traceability, Notre Dame Journal of Formal Logic, vol. 51 (2010), pp. 279290.
[9] Falconer, K., Fractal geometry: Mathematical foundations and applications, John Wiley & Sons, 2003.
[10] Fortnow, L., Hitchcock, J. M., Pavan, A., Vinochandran, N. V., and Wang, F., Extracting Kolmogorov complexity with applications to dimension zero-one laws, International Colloquium on Automata, Languages, and Programming (Bugliesi, M., Preneel, B., Sassone, V., and Wegener, I., editors), Lecture Notes in Computer Science, vol. 4051, Springer, Berlin, 2006, pp. 3545.
[11] Hausdorff, F., Dimension und äusseres Mass, Mathematische Annalen, vol. 79 (1919), pp. 157179.
[12] Lutz, J. H., The dimensions of individual strings and sequences, Information and Computation, vol. 84 (2003), pp. 4979.
[13] Mayordomo, E., A Kolmogorov complexity characterization of constructive Hausdorff dimension, Information Processing Letters, vol. 187 (2002), pp. 13.
[14] Miller, J. S., Extracting information is hard: a Turing degree of nonintegral effective Hausdorff dimension, Advances in Mathematics, vol. 226 (2011), pp. 373384.
[15] Nies, A. O., Computability and randomness, Oxford University Press, 2009.
[16] Soare, R. I., Recursively enumerable sets and degrees, Springer-Verlag, 1987.
[17] Sullivan, D., Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian group, Acta Mathematica, vol. 153 (1984), pp. 259277.
[18] Tricot, C., Two definitions of fractal dimension. Mathematical Proceedings of the Cambridge Philosophical Society, vol. 91 (1982), pp. 5774.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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