发表机构
CY Cergy Paris Universit\'e, Laboratoire Thema, 33 boulevard du port, F-95011 Cergy-Pontoise Cedex, France
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明平面紧集平行集空洞数的积分贝蒂数界，并构造反例说明固定半径下该界失效，为 Wiener 香肠持续同态极限定理提供确定性基础。
AI 中文摘要
设 A 为平面的非空紧子集，A(r) 为其距离 r 处的平行集。我们证明 A(r) 的空洞数（即其补集的有界连通分量数，也就是其一阶贝蒂数）满足对任意 r0 > 0，有 ∫₀^{r0} β₁(A(r)) dr ≤ 4050 (diam A)⁴ r₀⁻³，且被积函数在 r ≥ diam A / √3 时为零。证明依赖于 Fu 的定理（平面紧集距离函数的临界值构成零半维 Hausdorff 测度集）、Rataj、Spodarev 和 Meschenmoser 的两个引理，以及我们给出完整证明的临界值集间隙的平方根可和性估计。我们通过一个显式曲线族（两把相对的梳子）表明，在固定半径下不存在类似的界：一条具有有界长度、直径、振荡次数、平行集面积和平行集周长的连通曲线，可以在某一半径处具有任意多的空洞，因此积分估计不能由这些粗几何量在固定半径下的界所替代。我们还用距离函数局部极大值分量的数量来一致地界定空洞数随半径的变化，并记录了平行集边界长度由其面积界定的结果。这些结果为 Wiener 香肠持续同态的极限定理提供了确定性输入。
英文摘要
Let A be a nonempty compact subset of the plane and let A (r) be its parallel set at distance r. We prove that the number of holes of A (r) the number of bounded components of its complement, which is its rst Betti number satises $\infty$ r 0 $β$1(A (r) ) dr $\le$ 4050 (diam A) 4 r -3 0 for every r0 > 0, the integrand vanishing for r $\ge$ diam A/ $\sqrt$ 3. The proof rests on Fu's theorem that the critical values of the distance function of a planar compact set form a set of vanishing half-dimensional Hausdor measure, on two lemmas of Rataj, Spodarev and Meschenmoser, and on a square-root summability estimate for the gaps of the critical-value set, of which we give a complete proof. We show by an explicit family of curves two combs facing each other that no analogous bound can hold at a xed radius: a connected curve of bounded length, diameter, oscillation count, parallel-set area and parallel-set perimeter can have arbitrarily many holes at one radius, so the integrated estimate cannot be replaced by a xed-radius bound in terms of these coarse geometric quantities. We also bound the hole count uniformly in the radius by the number of components of local maxima of the distance function, and record a bound on the boundary length of a parallel set by its area. The results supply the deterministic input for limit theorems on the persistent homology of the Wiener sausage.
