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

Recently, the Dimension Problem for effective Hausdorff dimension was solved by J. Miller in [Mil], 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 [FHP06, BDS09, DH] 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).

[1]  C. Tricot.  Two definitions of fractional dimension , 1982, Mathematical Proceedings of the Cambridge Philosophical Society.

[2]  A. Nies.  Computability and randomness , 2009 .

[3]  Frank Stephan,et al.  Constructive Dimension and Turing Degrees , 2009, Theory of Computing Systems.

[4]  Chris J. Conidis.  Effective packing dimension of Π⁰₁-classes , 2008 .

[5]  Kenneth Falconer,et al.  Fractal Geometry: Mathematical Foundations and Applications , 1990 .

[6]  Terence Tao,et al.  A sum-product estimate in finite fields, and applications , 2003, math/0301343.

[7]  Avi Wigderson,et al.  Extracting randomness using few independent sources , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[8]  Rodney G. Downey,et al.  Effective Packing Dimension and Traceability , 2010, Notre Dame J. Formal Log..

[9]  Lance Fortnow,et al.  Extracting Kolmogorov complexity with applications to dimension zero-one laws , 2006, Inf. Comput..

[10]  R. Soare.  Recursively enumerable sets and degrees , 1987 .

[11]  D. Sullivan.  Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups , 1984 .

[12]  Joseph S. Miller.  Extracting information is hard: A Turing degree of non-integral effective Hausdorff dimension , 2011 .

[13]  F. Hausdorff.  Dimension und äußeres Maß , 1918 .

[14]  Jack H. Lutz,et al.  The dimensions of individual strings and sequences , 2002, Inf. Comput..

[15]  Elvira Mayordomo,et al.  A Kolmogorov complexity characterization of constructive Hausdorff dimension , 2002, Inf. Process. Lett..

[16]  Rodney G. Downey,et al.  Turing degrees of reals of positive effective packing dimension , 2008, Inf. Process. Lett..

[17]  Jack H. Lutz,et al.  Effective Strong Dimension, Algorithmic Information, and Computational Complexity , 2002, ArXiv.

[18]  Chris J. Conidis.  EFFECTIVE PACKING DIMENSION OF Π 01 -CLASSES , 2007 .

[19]  Rodney G. Downey,et al.  Algorithmic Randomness and Complexity , 2010, Theory and Applications of Computability.