## Effective packing dimension of Π 0 1-classes (2007)

Citations: | 2 - 1 self |

### BibTeX

@MISC{Conidis07effectivepacking,

author = {Chris J. Conidis},

title = {Effective packing dimension of Π 0 1-classes},

year = {2007}

}

### OpenURL

### Abstract

We construct a Π0 1-class X that has classical packing dimension 0 and effective packing dimension 1. This implies that, unlike in the case of effective Hausdorff dimension, there is no natural correspondence principle (as defined by Lutz) for effective packing dimension. We also examine the relationship between upper box dimension and packing A major theme of computability theory is the effectivization of classical mathematics. To do this one takes an existing (i.e. classical) mathematical notion and develops a new computability-theoretic analogue of that notion. Afterwards, one tries to determine the similarities and differences between the

### Citations

1736 | An introduction to Kolmogorov complexity and its applications
- Li, Vitányi
- 1993
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Citation Context ... ∈ A : | σ| < k}. We denote the plain and prefix-free Kolmogorov complexity of a string σ ∈ 2 <ω by C(σ) and K(σ), respectively. For more information on plain and prefixfree Kolmogorov complexity see =-=[6]-=-. A set X ⊆ 2 ω is a Π 0 1-class if there is a computable tree T ⊆ 2 <ω such that X is the set of paths through T . 2.2 Packing dimension In this section we define the notion of classical packing dime... |

552 |
Fractal Geometry: Mathematical Foundations and Applications
- Falconer
- 1990
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Citation Context ...T ⊆ 2 <ω such that X is the set of paths through T . 2.2 Packing dimension In this section we define the notion of classical packing dimension. For more information on classical packing dimension see =-=[4]-=-. For every k ∈ ω let Ak be the collection of prefix-free sets A ⊆ 2 <ω such that A<k = ∅. For every X ⊆ 2ω we now define � Ak(X) = A ∈ Ak : X ⊆ � C α � , α∈A Bk(X) = {A ∈ Ak : (∀α ∈ A)[C α ∩ X �= ∅]}... |

163 |
Algorithmic Randomness and Complexity
- Downey, Hirshfeldt
- 2009
(Show Context)
Citation Context ...ws that there is a Π 0 1-class X that has effective packing dimension 0 and upper box dimension 1. For further information on computability theory, effective randomness, and dimension theory, consult =-=[2, 3, 8, 9]-=-. 2 Definitions and notation 2.1 Cantor space and Π 0 1-classes In this article ω denotes the set of natural numbers, 2 <ω denotes the set of finite binary sequences, and 2 ω denotes the set of infini... |

103 | Computability and Randomness
- Nies
(Show Context)
Citation Context ...ws that there is a Π 0 1-class X that has effective packing dimension 0 and upper box dimension 1. For further information on computability theory, effective randomness, and dimension theory, consult =-=[2, 3, 8, 9]-=-. 2 Definitions and notation 2.1 Cantor space and Π 0 1-classes In this article ω denotes the set of natural numbers, 2 <ω denotes the set of finite binary sequences, and 2 ω denotes the set of infini... |

94 | The dimensions of individual strings and sequences
- Lutz
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Citation Context ...s R. Hirschfeldt. The author would also like to thank the American Institute of Mathematics for hosting a valuable workshop in effective randomness which lead to the publication of this article. 1sIn =-=[7]-=- Lutz effectivized the notion of Hausdorff dimension to obtain the notion of effective Hausdorff dimension. Furthermore, he conjectured that for Hausdorff dimension there is a correspondence principle... |

79 | Effective strong dimension, algorithmic information and computational complexity
- Athreya, Hitchcock, et al.
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Citation Context ...etermining its effective Hausdorff dimension, which, as is shown in [7], is the supremum of the effective Hausdorff dimensions of its individual points. Later, Athreya, Hitchcock, Lutz, and Mayordomo =-=[1]-=- effectivized the classical notion of packing dimension to obtain the notion of effective packing dimension. They also wondered whether or not there existed a correspondence principle for this new not... |

61 | Calibrating randomness
- Downey, Hirschfeldt, et al.
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Citation Context ...ws that there is a Π 0 1-class X that has effective packing dimension 0 and upper box dimension 1. For further information on computability theory, effective randomness, and dimension theory, consult =-=[2, 3, 8, 9]-=-. 2 Definitions and notation 2.1 Cantor space and Π 0 1-classes In this article ω denotes the set of natural numbers, 2 <ω denotes the set of finite binary sequences, and 2 ω denotes the set of infini... |

32 |
Randomness and computability: Open questions
- Nies, Miller
- 2006
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Citation Context |

24 |
Soare, Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets
- I
- 1987
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Citation Context .... Hitchcock [5] found such a class by showing that if X is a union of Π 0 1-classes, then the classical and effective Hausdorff dimensions of X are the same (for more information on Π 0 1-classes see =-=[10, 11]-=-). This is a beautiful and useful result, because it allows one to compute the classical Hausdorff dimension of a set by determining its effective Hausdorff dimension, which, as is shown in [7], is th... |

19 | Correspondence principles for effective dimensions. Theory of Computing Systems. To appear. Preliminary version appeared
- Hitchcock
- 2002
(Show Context)
Citation Context ...respondence principle. By correspondence principle we mean a theorem which says that there is a certain (natural) class of sets whose classical and effective Hausdorff dimensions are equal. Hitchcock =-=[5]-=- found such a class by showing that if X is a union of Π 0 1-classes, then the classical and effective Hausdorff dimensions of X are the same (for more information on Π 0 1-classes see [10, 11]). This... |

13 |
Soare, Computability Theory and Applications
- I
(Show Context)
Citation Context .... Hitchcock [5] found such a class by showing that if X is a union of Π 0 1-classes, then the classical and effective Hausdorff dimensions of X are the same (for more information on Π 0 1-classes see =-=[10, 11]-=-). This is a beautiful and useful result, because it allows one to compute the classical Hausdorff dimension of a set by determining its effective Hausdorff dimension, which, as is shown in [7], is th... |