On Falconer's distance set problem in the plane

Abstract

If $E\subset {\mathbb{R}}^2$ is a compact set of Hausdorff dimension greater than $5/4$, we prove that there is a point $x\in E$ so that the set of distances $\{|x-y|{\}}_{y\in E}$ has positive Lebesgue measure.

39 pages, 1 figure