A quantitative form of the planar Falconer distance theorem

We give a quantitative planar form of a multiscale argument for Falconer’s distance problem. For every $0<\eta <1$, we obtain a positive lower bound for the Lebesgue measure of the distance set of a compact subset of the unit ball supporting a probability measure with $\nu (B(x,r))\le 2r^{1+\eta }$. The bound depends only on $\eta$. For a compact set of diameter $D$ carrying a probability measure with $\mu (B(x,r))\le A(r\mathrm{/}D{)}^{1+\epsilon }$, localization gives $\mathrm{\mid }\mathrm{\Delta }E\mathrm{\mid }\ge c_{\epsilon ,p}D{A}^{-p}$ for every fixed $p>1\mathrm{/}\epsilon$. In particular, compact planar sets of Hausdorff dimension greater than one have distance sets of positive Lebesgue measure. The proof develops the multiscale method of the September 2026 manuscript of OpenAI, using one sequence of restricted distance measures for both the mass and Fourier estimates. A reconstruction lemma combines summable total-variation errors with summable squared Fourier norms. No endpoint or sharpness assertion is made.