Some connections between Falconer's distance set conjecture, and sets of Furstenburg type

TL;DR

This paper explores the deep connections between Falconer's distance set conjecture, Furstenburg sets, and Erdos's ring conjecture, showing their discretized versions are equivalent and emphasizing the importance of sub-ring structures.

Contribution

It establishes the equivalence of discretized versions of three major conjectures in geometric combinatorics, highlighting the role of sub-ring structures in progress.

Findings

Discretized versions of the conjectures are equivalent.

Progress requires proving the existence of 1/2-dimensional sub-rings.

Connections suggest new approaches to longstanding problems.

Abstract

In this paper we investigate three unsolved conjectures in geometric combinatorics, namely Falconer's distance set conjecture, the dimension of Furstenburg sets, and Erdos's ring conjecture. We formulate natural $\delta$-discretized versions of these conjectures and show that in a certain sense that these discretized versions are equivalent. In particular, it appears that to progress on any of these problems one must prove a quantitative statement about the existence of sub-rings of $R$ of dimension 1/2.