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

Abstract

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. Moreover, we are able to beat the $5/4$ 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 $5/4$ 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.