AI 中文总结
本文针对1-可求长集建立广义曲线投影的有限优见证原理，通过内蕴论证和新的折叠非退化条件得到确定性见证子与例外参数结构界，并将框架应用于多类非线性投影族，补充了分形集相关的例外集估计工作。
AI 中文摘要
对于正长度的1-可求长集合$E \subset \mathbb{R}^d$，Federer的一个定理表明，在$E$的任意$d$个线性无关正交投影中，至少有一个具有正长度。我们针对广义曲线投影建立了该有限见证原理的一个版本。给定标量值映射$φ_1,\ldots,φ_d:\mathbb{R}^d\longrightarrow\mathbb{R}$，我们引入典范编码映射$\mathsf{H}:=(φ_1,\ldots,φ_d)$。在$D\mathsf{H}$可逆的区域，通过局部双利普希茨变量替换和Federer定理可证，存在某个$j$使得$φ_j(E)$具有正长度。若$E$的一个正长度部分位于临界集中，只要$\mathsf{H}$在该处保持满切秩，我们转而在包含该部分的$C^1$超曲面上进行内蕴论证。这得到了大量确定性有限见证子，以及优见证性质不成立的例外参数的结构界。一个新的折叠非退化条件使得我们的构造在基础参数扰动下保持稳定。在可求长这一端点情形下，这些结论补充了Peres–Schlag的工作，后者针对Hausdorff维数严格大于1的分形集建立了例外集估计。随后我们将该框架应用于若干非线性投影族。仿射无关的固定平方距离映射满足切秩条件和折叠条件，而两个平面径向投影在其临界线上会损失切秩。我们还验证了更多特殊例子的假设条件，包括非线性各向异性距离、由多面体范数的光滑逼近产生的Bregman泛函，以及临界超曲面具有指定$C^2$几何结构的投影族。
英文摘要
For a $1$-rectifiable set $E \subset \mathbb{R}^d$ of positive length, a theorem of Federer shows that, among any $d$ linearly independent orthogonal projections of $E$, at least one has positive length. We develop a version of this finite-witness principle for generalized curve projections. Given scalar-valued mappings $φ_1,\ldots,φ_d:\mathbb{R}^d\longrightarrow\mathbb{R}$, we introduce the canonical encoding map $\mathsf{H}:=(φ_1,\ldots,φ_d)$. Where $D\mathsf{H}$ is invertible, a local bilipschitz change of variables and Federer's theorem show that $φ_j(E)$ has positive length for some $j$. If a positive-length portion of $E$ lies in the critical set, we instead argue intrinsically on a $C^1$ hypersurface containing it, provided that $\mathsf{H}$ retains full tangential rank there. This yields an abundance of deterministic finite witnesses, as well as structural bounds for those exceptional parameters where the good witness property fails. A new fold non-degeneracy condition makes our constructions stable under perturbations of the underlying parameters. At this rectifiable endpoint, these conclusions complement work of Peres--Schlag, which developed exceptional set estimates for fractal sets of Hausdorff dimension strictly greater than one. We then apply our framework to several nonlinear projection families. Affinely independent pinned squared-distance maps satisfy the tangential and fold conditions, whereas two planar radial projections lose tangential rank along their critical line. We also verify the hypotheses for more exotic examples, including nonlinear anisotropic distances, Bregman functionals arising from smooth approximations of polyhedral norms, and families whose critical hypersurfaces have prescribed $C^2$ geometry.
Comments47 pages. No figures. Comments welcome!
