  We survey the history and recent developments around two decades-old problems
that continue to attract a great deal of interest: the slicing $\times 2$,
$\times 3$ conjecture of H. Furstenberg in ergodic theory, and the distance set
problem in geometric measure theory introduced by K. Falconer. We discuss some
of the ideas behind our solution of Furstenberg's slicing conjecture, and
recent progress in Falconer's problem. While these two problems are on the
surface rather different, we emphasize some common themes in our approach:
analyzing fractals through a combinatorial description in terms of ``branching
numbers'', and viewing the problems through a ``multiscale projection'' lens.
