法瓦尔长度与广义投影
Favard length and generalized projections
AI总结：
该研究建立了广义投影与正交投影的局部可比性，推导了法瓦尔长度上界向非线性投影族的推广结论，还得到了纯不可约自相似1-集对应的圆并集及曲线并集的测度估计。
AI中文摘要：
我们研究与光滑非线性投影族相关的广义法瓦尔长度。在合适的正则性与横截性假设下，我们证明广义投影在足够小的尺度上与正交投影局部可比，这一结果给出了一个比较原理，可将经典法瓦尔长度的定量上界推广到广泛类别的非线性投影族。由此，纯不可约自相似1-集的法瓦尔长度已知的上界，可得到其对应广义法瓦尔长度的上界。我们还证明，当半径变化足够缓慢时，以纯不可约自相似1-集中的点为中心的圆的并集具有零勒贝格测度；更一般地，相同方法可得到来自合适水平集族的曲线并集的测度估计。
英文摘要：
We investigate generalized Favard lengths associated to smooth families of nonlinear projections. Under suitable regularity and transversality assumptions, we prove that generalized projections are locally comparable to orthogonal projections on sufficiently small scales. This yields a comparison principle that transfers quantitative upper bounds for classical Favard length to broad classes of nonlinear projection families. As a consequence, known upper bounds for the Favard length of purely unrectifiable self-similar 1-sets yield corresponding upper bounds for their generalized Favard lengths. We also prove that the union of circles with centers in a purely unrectifiable self-similar 1-set has Lebesgue measure zero whenever the radii vary sufficiently slowly. More generally, the same method yields measure estimates for unions of curves arising from suitable level-set families.
