## TURING DEGREES OF REALS OF POSITIVE EFFECTIVE PACKING DIMENSION

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@MISC{Downey_turingdegrees,

author = {Rod Downey and Noam Greenberg},

title = {TURING DEGREES OF REALS OF POSITIVE EFFECTIVE PACKING DIMENSION},

year = {}

}

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### Abstract

Abstract. 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 ≤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. 1.

### Citations

1791 | An introduction to Kolmogorov complexity and its applications 2nd edition - Li, Vitanyi - 1997 |

614 |
Fractal Geometry: Mathematical Foundations and Applications
- Falconer
- 1990
(Show Context)
Citation Context ...ctive packing dimension, which is an inner measure version, based around packing with balls of shrinking radius, of the notion of effective dimension, effectivizing the definition from, say, Falconer =-=[9]-=-. Again we will use a characterization as our working definition of effective packing dimension. This is a characterization due to Lutz [17]: the effective packing dimension of a real A ∈ 2 ω is He we... |

505 |
Recursively Enumerable Sets and Degrees
- Soare
- 1987
(Show Context)
Citation Context ...ons of Hausdorff dimension must agree on such classes. Our result shows that Conidis’ example can be found below any c.e., array noncomputable degree. Notation is standard and generally follows Soare =-=[23]-=-. For more on dimension and related notions of Kolmogorov complexity, the reader should also see Downey, Hirschfeldt, Nies and Terwijn [6], Downey and Hirschfeldt [5] or Nies [20]. 2. Clumpy trees We ... |

173 |
Algorithmic randomness and complexity
- Downey, Hirschfeldt
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Citation Context ...packing dimension 1, such that for all reals B �T A, B is not random. The corollary follows since it is well known that no random real can have minimal Turing degree. (See e.g. Downey and Hirschfeldt =-=[5]-=- or Nies [20].) Theorem 1.1 raises a more general question: what kind of Turing degrees contain reals with positive effective packing dimension? we note that having effective packing dimension one is ... |

113 | Computability and randomness
- Nies
- 2009
(Show Context)
Citation Context ...nsion 1, such that for all reals B �T A, B is not random. The corollary follows since it is well known that no random real can have minimal Turing degree. (See e.g. Downey and Hirschfeldt [5] or Nies =-=[20]-=-.) Theorem 1.1 raises a more general question: what kind of Turing degrees contain reals with positive effective packing dimension? we note that having effective packing dimension one is a property th... |

93 | The dimensions of individual strings and sequences
- Lutz
- 2002
(Show Context)
Citation Context ...mme has been around since the 1950’s beginning with the work of de Leeuw, Moore, Shannon and Shapiro [5]. A new initiative in the study of algorithmic randomness was the work of Lutz and others (e.g. =-=[1, 18, 19]-=-, Staiger’s [27]) who effectivized the refinements of the notion of Lebesgue measure known as dimensions. The best known of these is the notion of Hausdorff dimension [13]. For our purposes, we will t... |

74 | Effective strong dimension, algorithmic information, and computational complexity
- Athreya, Hitchcock, et al.
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Citation Context ... 1950’s beginning with the work of de Leeuw, Moore, Shannon and Shapiro [4]. A new initiative in the study of algorithmic randomness was the work of Staiger, Lutz and his co-authors, and others (e.g. =-=[1, 16, 17, 24]-=-) who effectivized the refinements of the notion of Lebesgue measure known as dimensions. The best known of these is the notion of Hausdorff dimension [11]. For our purposes, we will take as our defin... |

71 |
A Kolmogorov complexity characterization of constructive Hausdorff dimension
- Mayordomo
- 2002
(Show Context)
Citation Context ...ue measure known as dimensions. The best known of these is the notion of Hausdorff dimension [11]. For our purposes, we will take as our definition for this notion a characterization due to Mayordomo =-=[18]-=- which is that the effective Hausdorff dimension of a real A ∈ 2ω is K(A ↾ n) dim(A) = lim inf . n→∞ n Here K denotes prefix-free Kolmogorov complexity (though plain complexity C would be okay as well... |

68 | Category and measure in complexity classes
- Lutz
- 1990
(Show Context)
Citation Context ... 1950’s beginning with the work of de Leeuw, Moore, Shannon and Shapiro [4]. A new initiative in the study of algorithmic randomness was the work of Staiger, Lutz and his co-authors, and others (e.g. =-=[1, 16, 17, 24]-=-) who effectivized the refinements of the notion of Lebesgue measure known as dimensions. The best known of these is the notion of Hausdorff dimension [11]. For our purposes, we will take as our defin... |

67 | Kolmogorov complexity and Hausdorff dimension
- Staiger
- 1993
(Show Context)
Citation Context ... 1950’s beginning with the work of de Leeuw, Moore, Shannon and Shapiro [4]. A new initiative in the study of algorithmic randomness was the work of Staiger, Lutz and his co-authors, and others (e.g. =-=[1, 16, 17, 24]-=-) who effectivized the refinements of the notion of Lebesgue measure known as dimensions. The best known of these is the notion of Hausdorff dimension [11]. For our purposes, we will take as our defin... |

65 | Calibrating randomness
- Downey, Hirschfeldt, et al.
(Show Context)
Citation Context ...Fund of New Zealand. 1s2 ROD DOWNEY AND NOAM GREENBERG If A has positive effective Hausdorff dimension, is there a random real B �T A? The reader should also see Downey, Hirschfeldt, Nies and Terwijn =-=[6]-=- and Miller and Nies [19]. There are of course many other notions of dimension, and hence of effective dimension, in the study of measure and effective measure. The broken dimension question has been ... |

31 |
Randomness and computability: open questions
- Miller, Nies
(Show Context)
Citation Context ... ROD DOWNEY AND NOAM GREENBERG If A has positive effective Hausdorff dimension, is there a random real B �T A? The reader should also see Downey, Hirschfeldt, Nies and Terwijn [6] and Miller and Nies =-=[19]-=-. There are of course many other notions of dimension, and hence of effective dimension, in the study of measure and effective measure. The broken dimension question has been open for all such dimensi... |

30 |
Array nonrecursive sets and multiple permitting arguments, Recursion Theory
- Downey, Jockusch, et al.
- 1990
(Show Context)
Citation Context ... that have positive effective Hausdorff measure has measure 1 but is meagre.) This is why our gaze turns naturally to the class of array noncomputable degrees, introduced by Downey, Jockusch and Stob =-=[7, 8]-=-. In the case of c.e. degrees, we get a complete characterisation. Recall that a Turing degree a is array noncomputable if for all f �wtt ∅ ′ , there is a function g �T a such that ∃ ∞ n(g(n) > f(n)).... |

26 | Array nonrecursive degrees and genericity
- Downey, Jockusch, et al.
(Show Context)
Citation Context ... that have positive effective Hausdorff measure has measure 1 but is meagre.) This is why our gaze turns naturally to the class of array noncomputable degrees, introduced by Downey, Jockusch and Stob =-=[7, 8]-=-. In the case of c.e. degrees, we get a complete characterisation. Recall that a Turing degree a is array noncomputable if for all f �wtt ∅ ′ , there is a function g �T a such that ∃ ∞ n(g(n) > f(n)).... |

25 | Kolmogorov complexity and instance complexity of recursively enumerable sets
- KUMMER
- 1996
(Show Context)
Citation Context ...� 2 log n + O(1) for infinitely many n. Kummer proved the following gap theorem. Recall that an order function is an unbounded and nondecreasing computable function. Theorem 1.3 (Kummer’s Gap Theorem =-=[13]-=-).sTURING DEGREES OF REALS OF POSITIVE EFFECTIVE PACKING DIMENSION 3 (i) A c.e., array noncomputable degree contains a c.e. set A such that C(A ↾ n) � 2 log n − O(1) for infinitely many n. (ii) If A h... |

21 | A lower cone in the wtt degrees of non-integral effective dimension
- Nies, Reimann
- 2006
(Show Context)
Citation Context ...imann himself and his coauthors. Reimann and Terwijn [24, 3.11] showed that the answer to the question is negative if Turing reducibility is replaced by many-one reducibility; later, Nies and Reimann =-=[23]-=- improved this result to the weak truth-table degrees. Indeed, both results established lower cones of degrees of fractional dimension. Continuing in this line, Bienvenu, Doty and Stephan [3] showed t... |

21 |
Computability and Fractal Dimension
- Reimann
- 2004
(Show Context)
Citation Context ...ctively reversed, which would mean that we could extract random information from every real of positive effective Hausdorff dimension. This question was first articulated by Jan Reimann in his thesis =-=[24]-=- : The first author’s research was supported by the Marsden Fund of New Zealand. 12 ROD DOWNEY AND NOAM GREENBERG If A has positive effective Hausdorff dimension, is there a random real B �T A? The r... |

16 | Two sources are better than one for increasing the Kolmogorov complexity of infinite sequences. Theory Comput
- Zimand
(Show Context)
Citation Context ... hand, some positive results were also discovered. Doty [6] showed that arbitrarily high effective packing dimension (below 1) can be extracted from positive effective Hausdorff dimension; and Zimand =-=[30]-=- showed, in contrast with Bienvenu’s, Doty’s and Stephan’s result, that if one allows two independent (mutually incompressible) reals, then one can increase positive increase positive effective Hausdo... |

15 | Degrees of Unsolvability: Local and Global Theory - Lerman - 1983 |

15 | Effective fractal dimensions
- Lutz
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Citation Context |

13 |
Weak recursive degrees and a problem of Spector, in Recursion Theory and Complexity
- Ishmukhametov
- 1999
(Show Context)
Citation Context ....e. sequence 〈Tx〉 such that for all x, |Tx| � h(x) and f(x) ∈ Tx. All c.e. traceable degrees are array computable. C.e. traceability was introduced by Zambella [26] and independently by Ishmukhametov =-=[12]-=-, who noted that in the c.e. degrees, c.e. traceability and array computability coincided. Hence the second part of Kummer’s theorem follows from: Proposition 1.4. If A is c.e. traceable, then then fo... |

12 | Algorithmic randomness and lowness
- Terwijn, Zambella
(Show Context)
Citation Context ... This is similar to Kummer’s proof. We know that if A is c.e. traceable then any order function h can serve for a bound for the size of traces of functions computable from h (see Terwijn and Zambella =-=[25]-=-.) We trace the function n ↦→ A ↾ n. Let 〈Tn〉 serve as a trace for this function with bound h. Then to specify A ↾ n we need n (this takes log n many bits), a constant number of bits to describe the m... |

11 | Constructive dimension and weak truth-table degrees
- Bienvenu, Doty, et al.
(Show Context)
Citation Context ...Reimann [23] improved this result to the weak truth-table degrees. Indeed, both results established lower cones of degrees of fractional dimension. Continuing in this line, Bienvenu, Doty and Stephan =-=[3]-=- showed that there is no uniform way to extract randomness from positive effective Hausdorff dimension (using full Turing reducibility). Reimann and Slaman [24, 4.17] showed that the answer to Reimann... |

10 |
On Sequences with Simple Initial Segments, ILLC
- Zambella
- 1990
(Show Context)
Citation Context ...at for all f �T a there is a uniformly c.e. sequence 〈Tx〉 such that for all x, |Tx| � h(x) and f(x) ∈ Tx. All c.e. traceable degrees are array computable. C.e. traceability was introduced by Zambella =-=[26]-=- and independently by Ishmukhametov [12], who noted that in the c.e. degrees, c.e. traceability and array computability coincided. Hence the second part of Kummer’s theorem follows from: Proposition 1... |

8 |
Computability by probabilistic machines, Automata studies
- Leeuw, Moore, et al.
- 1956
(Show Context)
Citation Context ...nd their relationship with measures of computational complexity such as Turing degrees. This programme has been around since the 1950’s beginning with the work of de Leeuw, Moore, Shannon and Shapiro =-=[4]-=-. A new initiative in the study of algorithmic randomness was the work of Staiger, Lutz and his co-authors, and others (e.g. [1, 16, 17, 24]) who effectivized the refinements of the notion of Lebesgue... |

8 |
Dimension und äußeres Maß, Mathematische Annalen 79
- Hausdorff
- 1919
(Show Context)
Citation Context ... his co-authors, and others (e.g. [1, 16, 17, 24]) who effectivized the refinements of the notion of Lebesgue measure known as dimensions. The best known of these is the notion of Hausdorff dimension =-=[11]-=-. For our purposes, we will take as our definition for this notion a characterization due to Mayordomo [18] which is that the effective Hausdorff dimension of a real A ∈ 2ω is K(A ↾ n) dim(A) = lim in... |

8 | Dimension extractors and optimal decompression
- Doty
- 2007
(Show Context)
Citation Context ...e to some order function h; whereas a real has positive packing dimension if it is not null relative to some linear order function. On the other hand, some positive results were also discovered. Doty =-=[6]-=- showed that arbitrarily high effective packing dimension (below 1) can be extracted from positive effective Hausdorff dimension; and Zimand [30] showed, in contrast with Bienvenu’s, Doty’s and Stepha... |

7 |
Complexity of programs to determine whether natural numbers not greater than n belong to a recursively enumerable set
- Barzdins
- 1968
(Show Context)
Citation Context ...llection of apparently unrelated phenomena. Early on, it was realized that at least in the c.e. case, array noncomputability is intimately related to Kolmogorov complexity. An old result of Barzdins’ =-=[2]-=- states that if A is a c.e. set then C(A ↾ n) � 2 log n + O(1). In unpublished work Solovay (see [5]) showed that it is not possible for a c.e. set to have C(A ↾ n) � 2 log n + O(1) for all n. It was ... |

4 |
Forcing with perfect closed sets, 1971 Axiomatic Set Theory, vol
- Sacks
- 1967
(Show Context)
Citation Context ...eneric filter G ⊂ P yields a real XG ∈ 2 ω which has effective packing dimension 1 and minimal Turing degree. This is a modification of the standard forcing with computable perfect trees due to Sacks =-=[22]-=-. We need to restrict the kind of perfect trees we use so that we can always choose strings that are sufficiently complicated (i.e., not easily compressed), to be initial segments of the real we build... |

2 |
Computability and dimension. Unpublished notes
- Reimann
- 2004
(Show Context)
Citation Context ...ctively reversed, which would mean that we could extract random information from every real of positive effective Hausdorff dimension. This question was first articulated by Jan Reimann in his thesis =-=[21]-=- : The first author’s research was supported by the Marsden Fund of New Zealand. 1s2 ROD DOWNEY AND NOAM GREENBERG If A has positive effective Hausdorff dimension, is there a random real B �T A? The r... |

1 |
Double Jump Inversions and Strong Minimal Covers in the Turing Degrees
- Gabbay
- 2004
(Show Context)
Citation Context ...ontain a real whose effective packing dimension is positive. We note that this, together with Theorem 1.1, implies the fact that there is a minimal degree which is not c.e. traceable (see for example =-=[10]-=-). This shows that c.e. traceability is not sufficient to settle Yates’ question on strong minimal covers of minimal degrees (Ishmukhametov [12] showed that every c.e. traceable degree has a strong mi... |