Topics in computability theory: Boolean algebras and effective packing dimension

This thesis considers problems from two areas of computability theory, and, as such, is divided into two parts.;The first part is be the construction of a real number A ≤T 0' which has positive effective packing dimension, but which cannot compute any real number with effective packing dimension 1. The effective packing dimension of a real number is a measure of its complexity, and is a value in [0, 1]. If the effective packing dimension of a real number A lies in (0, 1), we may regard A as being "partially random". The real number which is constructed in this section shows a limitation on how much randomness can be computably extracted from a partially random source. This is an extension of a result of Conidis, who carried out a similar construction. However, Conidis' construction was not carried out below 0'.;The second part is part of an ongoing program to answer the low n problem for Boolean algebras. The lown problem asks whether every lown Boolean algebra is isomorphic to a computable Boolean algebra. The behaviour for n ≤ 4 suggests that not only do such isomorphisms always exist, but that in each case, 0(n+2) can compute one such isomorphism. Harris and Montalban constructed a low5 Boolean algebra which refutes the purported bound. This part of the thesis develops machinery which proves that any counterexample along the lines of that given by Harris and Montalban must occur as a member of an infinite family. In particular, using the method of Harris and Montalban, the machinery shows that for each n there is a low2n+5 Boolean algebra which is not isomorphic to any low2n+4 Boolean algebra via any 0(2n+7)-computable map.