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

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

2001-01-23

arxiv.org/abs/math/0101195v1

Mathematics

Classical Analysis and ODEs

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

Scientific paper

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.

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