The strength of the Besicovitch-Davies theorem

Bjørn Kjos-Hanssen, Jan Severin Reimann

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

1 Scopus citations


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
Number of pages10
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


Other6th Conference on Computability in Europe, CiE 2010
CityPonta Delgada, Azores

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)


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