We refine the genericity concept of [1], 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]. We also introduce strong generic density, and show that it is related to packing dimension [2]. 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 [3], to the packing dimension [2] setting, for k-bounded-truth-table reductions, under any (biased coin) probability measure.
展开▼
