Some connections between Falconer's distance set conjecture, and sets of Furstenburg type
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 subrings of $R$ of dimension 1/2.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 January 2001
 arXiv:
 arXiv:math/0101195
 Bibcode:
 2001math......1195H
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematics  Combinatorics;
 05B99;
 28A78;
 28A75
 EPrint:
 42 pages, 5 figures, submitted, New York Journal of Mathematics