12 research outputs found
Lowness notions, measure and domination
We show that positive measure domination implies uniform almost everywhere
domination and that this proof translates into a proof in the subsystem
WWKL (but not in RCA) of the equivalence of various Lebesgue measure
regularity statements introduced by Dobrinen and Simpson. This work also allows
us to prove that low for weak -randomness is the same as low for
Martin-L\"of randomness (a result independently obtained by Nies). Using the
same technique, we show that implies , generalizing the
fact that low for Martin-L\"of randomness implies low for
Characterizing the strongly jump-traceable sets via randomness
We show that if a set is computable from every superlow 1-random set,
then is strongly jump-traceable. This theorem shows that the computably
enumerable (c.e.) strongly jump-traceable sets are exactly the c.e.\ sets
computable from every superlow 1-random set.
We also prove the analogous result for superhighness: a c.e.\ set is strongly
jump-traceable if and only if it is computable from every superhigh 1-random
set.
Finally, we show that for each cost function with the limit condition
there is a 1-random set such that every c.e.\ set
obeys . To do so, we connect cost function strength and the strength of
randomness notions. This result gives a full correspondence between obedience
of cost functions and being computable from 1-random sets.Comment: 41 page
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