  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.
