### Abstract

A theorem of Besicovitch and Davies implies for Cantor space 2 ^{ω} that each Σ^{1}
_{1}(analytic) class of positive Hausdorff dimension contains a Π^{0}
_{1}(closed) subclass of positive dimension. We consider the weak (Muchnik) reducibility ≤_{w} in connection with the mass problem S(U) of computing a set X ⊆ ω such that the Σ^{1}
_{1} class U of positive dimension has a Π^{0}
_{1}(X) subclass of positive dimension. We determine the difficulty of the mass problems S(U) through the following results: (1) Y is hyperarithmetic if and only if {Y} ≤_{w} S(U) for some U; (2) there is a U such that if Y is hyperarithmetic, then {Y} ≤_{w} S(U); (3) if Y is Π^{1}
_{1}-complete then S(U) ≤_{w} {Y} for all U.

Original language | English (US) |
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Title of host publication | Programs, Proofs, Processes - 6th Conference on Computability in Europe, CiE 2010, Proceedings |

Pages | 229-238 |

Number of pages | 10 |

DOIs | |

State | Published - Jul 29 2010 |

Event | 6th Conference on Computability in Europe, CiE 2010 - Ponta Delgada, Azores, Portugal Duration: Jun 30 2010 → Jul 4 2010 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 6158 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 6th Conference on Computability in Europe, CiE 2010 |
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Country | Portugal |

City | Ponta Delgada, Azores |

Period | 6/30/10 → 7/4/10 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Programs, Proofs, Processes - 6th Conference on Computability in Europe, CiE 2010, Proceedings*(pp. 229-238). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6158 LNCS). https://doi.org/10.1007/978-3-642-13962-8_26

}

*Programs, Proofs, Processes - 6th Conference on Computability in Europe, CiE 2010, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 6158 LNCS, pp. 229-238, 6th Conference on Computability in Europe, CiE 2010, Ponta Delgada, Azores, Portugal, 6/30/10. https://doi.org/10.1007/978-3-642-13962-8_26

**The strength of the Besicovitch-Davies theorem.** / Kjos-Hanssen, Bjørn; Reimann, Jan Severin.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - The strength of the Besicovitch-Davies theorem

AU - Kjos-Hanssen, Bjørn

AU - Reimann, Jan Severin

PY - 2010/7/29

Y1 - 2010/7/29

N2 - A theorem of Besicovitch and Davies implies for Cantor space 2 ω that each Σ1 1(analytic) class of positive Hausdorff dimension contains a Π0 1(closed) subclass of positive dimension. We consider the weak (Muchnik) reducibility ≤w in connection with the mass problem S(U) of computing a set X ⊆ ω such that the Σ1 1 class U of positive dimension has a Π0 1(X) subclass of positive dimension. We determine the difficulty of the mass problems S(U) through the following results: (1) Y is hyperarithmetic if and only if {Y} ≤w S(U) for some U; (2) there is a U such that if Y is hyperarithmetic, then {Y} ≤w S(U); (3) if Y is Π1 1-complete then S(U) ≤w {Y} for all U.

AB - A theorem of Besicovitch and Davies implies for Cantor space 2 ω that each Σ1 1(analytic) class of positive Hausdorff dimension contains a Π0 1(closed) subclass of positive dimension. We consider the weak (Muchnik) reducibility ≤w in connection with the mass problem S(U) of computing a set X ⊆ ω such that the Σ1 1 class U of positive dimension has a Π0 1(X) subclass of positive dimension. We determine the difficulty of the mass problems S(U) through the following results: (1) Y is hyperarithmetic if and only if {Y} ≤w S(U) for some U; (2) there is a U such that if Y is hyperarithmetic, then {Y} ≤w S(U); (3) if Y is Π1 1-complete then S(U) ≤w {Y} for all U.

UR - http://www.scopus.com/inward/record.url?scp=77954879244&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77954879244&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-13962-8_26

DO - 10.1007/978-3-642-13962-8_26

M3 - Conference contribution

AN - SCOPUS:77954879244

SN - 3642139612

SN - 9783642139611

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 229

EP - 238

BT - Programs, Proofs, Processes - 6th Conference on Computability in Europe, CiE 2010, Proceedings

ER -