Characterizing the strongly jump-traceable sets via randomness

We show that if a set $A$ is computable from every superlow 1-random set, then $A$ 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 $c$ with the limit condition there is a 1-random $\Delta^0_2$ set $Y$ such that every c.e.\ set $A \le_T Y$ obeys $c$. 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 $\Delta^0_2$ 1-random sets.Comment: 41 page

Finding subsets of positive measure

An important theorem of geometric measure theory (first proved by Besicovitch and Davies for Euclidean space) says that every analytic set of non-zero $s$-dimensional Hausdorff measure $\mathcal H^s$ contains a closed subset of non-zero (and indeed finite) $\mathcal H^s$-measure. We investigate the question how hard it is to find such a set, in terms of the index set complexity, and in terms of the complexity of the parameter needed to define such a closed set. Among other results, we show that given a (lightface) $\Sigma^1_1$ set of reals in Cantor space, there is always a $\Pi^0_1(\mathcal{O})$ subset on non-zero $\mathcal H^s$-measure definable from Kleene's $\mathcal O$. On the other hand, there are $\Pi^0_2$ sets of reals where no hyperarithmetic real can define a closed subset of non-zero measure.Comment: This is an extended journal version of the conference paper "The Strength of the Besicovitch--Davies Theorem". The final publication of that paper is available at Springer via http://dx.doi.org/10.1007/978-3-642-13962-8_2