  The coarse similarity class $[A]$ of $A$ is the set of all $B$ whose
symmetric difference with $A$ has asymptotic density 0. There is a natural
metric $\delta$ on the space $\mathcal{S}$ of coarse similarity classes defined
by letting $\delta([A],[B])$ be the upper density of the symmetric difference
of $A$ and $B$. We study the resulting metric space, showing in particular that
between any two distinct points there are continuum many geodesic paths. We
also study subspaces of the form $\{[A] : A \in \mathcal U\}$ where $\mathcal
U$ is closed under Turing equivalence, and show that there is a tight
connection between topological properties of such a space and
computability-theoretic properties of $\mathcal U$.
  We then define a distance between Turing degrees based on Hausdorff distance
in this metric space. We adapt a proof of Monin to show that the distances
between degrees that occur are exactly 0, 1/2, and 1, and study which of these
values occur most frequently in the senses of measure and category. We define a
degree to be attractive if the class of all degrees at distance 1/2 from it has
measure 1, and dispersive otherwise. We study the distribution of attractive
and dispersive degrees. We also study some properties of the metric space of
Turing degrees under this Hausdorff distance, in particular the question of
which countable metric spaces are isometrically embeddable in it, giving a
graph-theoretic sufficient condition.
  We also study the computability-theoretic and reverse-mathematical aspects of
a Ramsey-theoretic theorem due to Mycielski, which in particular implies that
there is a perfect set whose elements are mutually 1-random, as well as a
perfect set whose elements are mutually 1-generic.
  Finally, we study the completeness of $(\mathcal S,\delta)$ from the
perspectives of computability theory and reverse mathematics.
