Improved dimension bounds for Falconer's distance problem in the plane

Abstract

Let $E\subset {\mathbb{R}}^2$ be analytic with $t={\dim }_{\mathrm{H}}E>1$. We prove ${\dim }_{\mathrm{H}}\Delta E\ge 4/5+{\eta }_0$ for an absolute, non-explicit constant ${\eta }_0>0$, and \[ \dim_{\mathrm H}ΔE\ge \frac{t^2-2t+5}{2t^2-7t+10} \quad\text{for }1<t\le\frac{11-\sqrt{41}}{4}, \] which reaches $7/8$ at the right endpoint. For pinned distances, ${\dim }_{\mathrm{H}}{\Delta }_{x}E\ge q_{\ast }=0.7813\dots$ for every $x\in {\mathbb{R}}^2$ outside an exceptional set of Hausdorff dimension at most one, with stronger bounds depending on $t$ and further exceptional-set estimates. If $E$ is compact and supports an $s$-Frostman measure $\mu$, $s>1$, then \[ \underline{\dim}_{\mathrm B}Δ_xE\ge \max\left\{\frac45+η_0,\min\left\{\frac{3s+1}{5},\frac78\right\}\right\} \] for $\mu$-almost every $x$. The same bound holds for pinned Hausdorff dimension under a common dyadic mass law, allowing oscillation with scale. We retain information about both endpoints in the Kolmogorov complexity estimates of Stull and Fiedler--Stull. Applying the partition estimates with the endpoints reversed yields $4/5$ and the rational bound. Equality forces both information profiles to have excess information on a common interval of scales. Encoding one source measure for all pins converts this constraint into a configuration of points and circles with many rich perpendicular bisectors. Lower bounds of Katz--Tardos and Ren--Wang contradict the upper bound of Fu--Gan--Ren; compactness then gives ${\eta }_0>0$. This rigidity argument has a precedent in the work of Héra, Shmerkin and Yavicoli on Furstenberg sets, using Bourgain's projection theorem. A separate finite-scale selection proves the pinned lower box bound.

133 pages