The strength of the Besicovitch-Davies theorem

Bjørn Kjos-Hanssen, Jan Severin Reimann

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Citation (Scopus)

Abstract

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.

Original languageEnglish (US)
Title of host publicationPrograms, Proofs, Processes - 6th Conference on Computability in Europe, CiE 2010, Proceedings
Pages229-238
Number of pages10
DOIs
StatePublished - Jul 29 2010
Event6th Conference on Computability in Europe, CiE 2010 - Ponta Delgada, Azores, Portugal
Duration: Jun 30 2010Jul 4 2010

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume6158 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other6th Conference on Computability in Europe, CiE 2010
CountryPortugal
CityPonta Delgada, Azores
Period6/30/107/4/10

Fingerprint

Theorem
Cantor
Reducibility
Hausdorff Dimension
If and only if
Imply
Closed
Computing
Class

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Kjos-Hanssen, B., & Reimann, J. S. (2010). The strength of the Besicovitch-Davies theorem. In 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
Kjos-Hanssen, Bjørn ; Reimann, Jan Severin. / The strength of the Besicovitch-Davies theorem. Programs, Proofs, Processes - 6th Conference on Computability in Europe, CiE 2010, Proceedings. 2010. pp. 229-238 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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Kjos-Hanssen, B & Reimann, JS 2010, The strength of the Besicovitch-Davies theorem. in 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.

Programs, Proofs, Processes - 6th Conference on Computability in Europe, CiE 2010, Proceedings. 2010. p. 229-238 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6158 LNCS).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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Kjos-Hanssen B, Reimann JS. The strength of the Besicovitch-Davies theorem. In Programs, Proofs, Processes - 6th Conference on Computability in Europe, CiE 2010, Proceedings. 2010. p. 229-238. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-642-13962-8_26