The Strength of the Besicovitch-Davies Theorem

A theorem of Besicovitch and Davies implies for Cantor space 2ω 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.

• Type: Conference
• Year: 2010
• Where: CIE
• Authors: Bjørn Kjos-Hanssen, Jan Reimann
• Keywords: Applied Computing, CIE 2010, Mass Problems, Positive Dimension, Positive Hausdorff Dimension