Slices and distances: on two problems of Furstenberg and Falconer

TL;DR

This paper surveys two longstanding problems in mathematics, the Furstenberg slicing conjecture and Falconer's distance set problem, highlighting recent progress and common analytical themes involving fractals and multiscale projections.

Contribution

The authors present their solution to Furstenberg's slicing conjecture and discuss recent advances in Falconer's distance set problem, emphasizing unified analytical approaches.

Findings

Solution to Furstenberg's slicing conjecture presented.

Recent progress in Falconer's distance set problem discussed.

Common themes in fractal analysis and multiscale projections identified.

Abstract

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.