Theory of recursive functions and effective computability
Theory of recursive functions and effective computability
Higher recursion theory
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A theorem of Besicovitch and Davies implies for Cantor space 2ω that each Σ11 (analytic) class of positive Hausdorff dimension contains a Π10 (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 Σ11 class U of positive dimension has a Π10 (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 Π11 -complete then S(U) ≤w {Y} for all U.