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

We give an alternative proof of the $5/4$ result of Guth-Iosevich-Ou-Wang on the Falconer distance conjecture: if a compact set $E$ in the plane has Hausdorff dimension $>5/4$, then there exists $x\in E$ such that the pinned distance set $$Δ_x(E):=\{|x-y|: y\in E\} $$ has positive Lebesgue measure.