  We prove various results connected together by the common thread of
computability theory. First, we investigate a new notion of algorithmic
dimension, the inescapable dimension, which lies between the effective
Hausdorff and packing dimensions. We also study its generalizations, obtaining
an embedding of the Turing degrees into notions of dimension. We then
investigate a new notion of computability theoretic immunity that arose in the
course of the previous study, that of a set of natural numbers with no
co-enumerable subsets. We demonstrate how this notion of $\Pi^0_1$-immunity is
connected to other immunity notions, and construct $\Pi^0_1$-immune reals
throughout the high/low and Ershov hierarchies. We also study those degrees
that cannot compute or cannot co-enumerate a $\Pi^0_1$-immune set. Finally, we
discuss a recently discovered truth-table reduction for transforming a
Kolmogorov--Loveland random input into a Martin-L\"of random output by
exploiting the fact that at least one half of such a KL-random is itself
ML-random. We show that there is no better algorithm relying on this fact,
i.e., there is no positive, linear, or bounded truth-table reduction which does
this. We also generalize these results to the problem of outputting randomness
from infinitely many inputs, only some of which are random.
