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

Published online by Cambridge University Press:  12 March 2014

Chris J. Conidis
Chris J. Conidis*
Affiliation:
Department of Pure Mathematics, University of Waterloo,Waterloo, ON N2L 3G1, Canada, E-mail: cconidis@math.uwaterloo.ca
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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).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 2 , June 2012 , pp. 447 - 474
Copyright
Copyright © Association for Symbolic Logic 2012

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References

REFERENCES

[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. 671–705.Google Scholar
[2] Barak, B., Impagliazzo, R., and Wigderson, A., Extracting randomness using few independent sources, SIAM Journal on Computing, vol. 36 (2006), pp. 1095–1118.CrossRefGoogle Scholar
[3] Bienvenu, L., Doty, D., and Stephan, F., Constructive dimension and Turing degrees, Theory of Computing Systems, vol. 45 (2009), pp. 740–755.Google Scholar
[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. 27–57.Google Scholar
[5] Conidis, C. J., Effective packing dimension of -classes, Proceedings of the American Mathematical Society, vol. 136 (2008), pp. 3655–3662.Google Scholar
[6] Downey, R. G. and Greenberg, N., Turing degrees of reals of positive packing dimension, Information Processing Letters, vol. 108 (2008), pp. 198–203.Google Scholar
[7] Downey, R. G. and Hirshfeldt, D. R., Algorithmic randomness and complexity, Springer-Verlag, 2010.CrossRefGoogle Scholar
[8] Downey, R. G. and Ng, K. M., Effective packing dimension and traceability, Notre Dame Journal of Formal Logic, vol. 51 (2010), pp. 279–290.Google Scholar
[9] Falconer, K., Fractal geometry: Mathematical foundations and applications, John Wiley & Sons, 2003.Google Scholar
[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. 35–45.Google Scholar
[11] Hausdorff, F., Dimension und äusseres Mass, Mathematische Annalen, vol. 79 (1919), pp. 157–179.Google Scholar
[12] Lutz, J. H., The dimensions of individual strings and sequences, Information and Computation, vol. 84 (2003), pp. 49–79.Google Scholar
[13] Mayordomo, E., A Kolmogorov complexity characterization of constructive Hausdorff dimension, Information Processing Letters, vol. 187 (2002), pp. 1–3.Google Scholar
[14] Miller, J. S., Extracting information is hard: a Turing degree of nonintegral effective Hausdorff dimension, Advances in Mathematics, vol. 226 (2011), pp. 373–384.Google Scholar
[15] Nies, A. O., Computability and randomness, Oxford University Press, 2009.Google Scholar
[16] Soare, R. I., Recursively enumerable sets and degrees, Springer-Verlag, 1987.Google Scholar
[17] Sullivan, D., Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian group, Acta Mathematica, vol. 153 (1984), pp. 259–277.Google Scholar
[18] Tricot, C., Two definitions of fractal dimension. Mathematical Proceedings of the Cambridge Philosophical Society, vol. 91 (1982), pp. 57–74.Google Scholar