Slices and distances: on two problems of Furstenberg and Falconer

Abstract

We survey the history and recent developments around two decades-old problems that continue to attract a great deal of interest: the slicing ×2, ×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.

A Survey on the Hausdorff Dimension of Intersections

“…which is a Frostman-type condition along the level sets of the P λ . With some further work, it leads to (35). The proof of Theorem 11 is similar.…”

“…In the plane, the result of [33] says that the distance set of A has a positive Lebesgue measure if dim A > 5/4. See Shmerkin's survey [35] for the distance set and related problems.…”

New estimates on the size of $(α,2α)$-Furstenberg sets

“…The next lemma says that every discretized set has a large subset that is Moran-regular. See [15,Lemma 3.4] or [19,Lemma 2.2] for a detailed proof. Lemma 3.7.…”