if $E \subset {\mathbb{R}}^2$ is a compact set with Hausdorff dimension greater than $4/3$, then $\Delta(E)$ has positive Lebesgue measure. In this paper, we improve the bound.

\[main1\] If $E \subset {\mathbb{R}}^2$ is a compact set with Hausdorff dimension greater than $5/4$, then $\Delta(E)$ has positive Lebesgue measure.

In higher dimensions, Erdoğan proved in [@Erd05] that if $dim_{{\mathcal H}}(E)>\frac{d}{2}+\frac{1}{3}$, then ${\mathcal L}(\Delta(E))>0$. Recently, these estimates were improved for all $d \ge 3$ by Du, Guth, Ou, Wang, Wilson, and Zhang [@DGOWWZ18]. In dimension 3, they showed that the Falconer conjecture holds when $dim_{{\mathcal{H}}}(E) > 9/5$. The estimates for $d \ge 4$ were further improved by Du and Zhang in [@DZ18]. For large $d$, they prove that Falconer’s conjecture holds when $dim_{{\mathcal{H}}}(E) > \frac{d}{2} + \frac{1}{4} + o(1)$. These works brought into play the decoupling theorem of Bourgain and Demeter [@BD15]. This approach will also play a key role in our proof.

Returning to the planar case, there have been a number of important recent results. Orponen [@O17] proved that if $E$ is a compact Ahlfors-David regular set of dimension $s \ge 1$, then $\Delta(E)$ has packing dimension 1. Note that packing dimension 1 is only slightly weaker than positive measure. 