On Falconer's distance set problem in the plane

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