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

Let $E\subset {\mathbb{R}}^2$ be analytic with $t={\dim }_{\mathrm{H}}E>1$. We prove ${\dim }_{\mathrm{H}}\mathrm{\Delta }E\ge 4\mathrm{/}5+{\eta }_0$ for an absolute, non-explicit constant ${\eta }_0>0$, and $${\dim }_{\mathrm{H}}\mathrm{\Delta }E\ge \frac{t^2-2t+5}{2t^2-7t+10}\text{for }1<t\le \frac{11-\sqrt{41}}{4},$$ which reaches $7\mathrm{/}8$ at the right endpoint. For pinned distances, ${\dim }_{\mathrm{H}}{\mathrm{\Delta }}_{x}E\ge q_{\ast }=0.7813\dots$ 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 $${\underset{}{\dim }}_{\mathrm{B}}{\mathrm{\Delta }}_{x}E\ge \max \{\frac{4}{5}+{\eta }_0,\min \{\frac{3s+1}{5},\frac{7}{8}\}\}$$ 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 $4\mathrm{/}5$ 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.