Generic Density and Small Span Theorem

Abstract

We refine the genericity concept of Ambos-Spies, by assigning a real number in [0, 1] to every generic set, called its generic density. We construct sets of generic density any E-computable real in [0,1], and show a relationship between generic density and Lutz resource bounded dimension. We also introduce strong generic density, and show that it is related to packing dimension. We show that all four notions are different. We show that whereas dimension notions depend on the underlying probability measure, generic density does not, which implies that every dimension result proved by generic density arguments, simultaneously holds under any (biased coin based) probability measure. We prove such a result: we improve the small span theorem of Juedes and Lutz, to the packing dimension setting, for k-bounded-truthtable reductions, under any (biased coin) probability measure.