  We discuss a number of naturally arising problems in arithmetic, culled from
completely unrelated sources, which turn out to have a common formulation
involving "thin" orbits. These include the local-global problem for integral
Apollonian gaskets and Zaremba's Conjecture on finite continued fractions with
absolutely bounded partial quotients. Though these problems could have been
posed by the ancient Greeks, recent progress comes from a pleasant synthesis of
modern techniques from a variety of fields, including harmonic analysis,
algebra, geometry, combinatorics, and dynamics. We describe the problems,
partial progress, and some of the tools alluded to above.
