AI 中文总结
本文改进了平面Falconer距离问题的维数下界，通过Kolmogorov复杂度与刚性论证得到绝对常数改进及依赖于维数的有理界，并推广到固定距离情形。
AI 中文摘要
设$E\subset\mathbb R^2$为解析集，$t=\dim_{\mathrm H}E>1$。我们证明$\dim_{\mathrm H}\Delta E\ge 4/5+\eta_0$，其中$\eta_0>0$为绝对非显式常数，并且当$1<t\le\frac{11-\sqrt{41}}{4}$时，$\dim_{\mathrm H}\Delta E\ge \frac{t^2-2t+5}{2t^2-7t+10}$，该界在右端点达到$7/8$。对于固定距离，对每个$\mathbb R^2$中除去一个Hausdorff维数至多为1的例外集之外的$x$，有$\dim_{\mathrm H}\Delta_xE\ge q_*=0.7813\ldots$，并且根据$t$有更强的界以及进一步的例外集估计。若$E$为紧集且支撑一个$s$-Frostman测度$\mu$，$s>1$，则对$\mu$-几乎每个$x$，有$\underline{\dim}_{\mathrm B}\Delta_xE\ge \max\left\{\frac45+\eta_0,\min\left\{\frac{3s+1}{5},\frac78\right\}\right\}$。在共同的二进质量定律下，同样的界也适用于固定Hausdorff维数，允许随尺度振荡。我们在Stull和Fiedler--Stull的Kolmogorov复杂度估计中保留了关于两个端点的信息。将端点反转的划分估计应用于$4/5$和有理数界。相等性迫使两个信息分布在共同的尺度区间上具有多余信息。对所有固定点编码一个源测度，将此约束转化为具有许多富垂直平分线的点和圆的配置。Katz--Tardos和Ren--Wang的下界与Fu--Gan--Ren的上界矛盾；紧性随后给出$\eta_0>0$。此刚性论证在Héra、Shmerkin和Yavicoli关于Furstenberg集的工作中有先例，使用了Bourgain的投影定理。一个独立的有限尺度选择证明了固定下盒维数界。
英文摘要
Let $E\subset\mathbb R^2$ be analytic with $t=\dim_{\mathrm H}E>1$. We prove $\dim_{\mathrm H}ΔE\ge 4/5+η_0$ for an absolute, non-explicit constant $η_0>0$, 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, $\dim_{\mathrm H}Δ_xE\ge q_*=0.7813\ldots$ 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 $μ$, $s>1$, then \[ \underline{\dim}_{\mathrm B}Δ_xE\ge \max\left\{\frac45+η_0,\min\left\{\frac{3s+1}{5},\frac78\right\}\right\} \] for $μ$-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 $η_0>0$. 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.
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