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

Slices and distances: on two problems of Furstenberg and Falconer