Some connections between Falconer's distance set conjecture, and sets of Furstenburg type (original) (raw)

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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\deltadelta-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 RRR of dimension 1/2.

Publication:

arXiv Mathematics e-prints

Pub Date:

January 2001

DOI:

10.48550/arXiv.math/0101195

arXiv:

arXiv:math/0101195

Bibcode:

2001math......1195H

Keywords:

• Mathematics - Classical Analysis and ODEs;
• Mathematics - Combinatorics;
• 05B99;
• 28A78;
• 28A75

E-Print:

42 pages, 5 figures, submitted, New York Journal of Mathematics