Lebesgue measure of distance sets and $L^2$ ball inflation

We give an alternative proof of the $5\mathrm{/}4$ result of Guth-Iosevich-Ou-Wang on the Falconer distance conjecture: if a compact set $E$ in the plane has Hausdorff dimension $>5\mathrm{/}4$, then there exists $x\in E$ such that the pinned distance set $${\mathrm{\Delta }}_x(E):=\{\mathrm{\mid }x-y\mathrm{\mid }:y\in E\}$$ has positive Lebesgue measure. Moreover, we are able to beat the $5\mathrm{/}4$ barrier if the packing dimension of $E$ is $<2$. More generally, we obtain sufficient conditions involving both the Hausdorff and packing dimensions of $E$. These conditions improve the $5\mathrm{/}4$ threshold when the packing dimension of $E$ is $<2$, and recover the second author’s result for ${\dim }_{H}E={\dim }_{P}E>1$. The proof is based on $L^2$ ball inflation and does not use decoupling inequalities.