标题： 粗略和点切向场 显示英文标题

标题： Coarse and pointwise tangent fields

摘要： 阿尔贝蒂、切尔尼亚伊和普莱斯为欧几里得空间中的子集引入了一个“点态（弱）切向量场”的概念——一个包含通过该集合的每条曲线的几乎每个切线的场——并证明了平面上所有面积为零的集合都具有一维切向量场。 我们从两个不同的方向扩展了他们的结果。 首先，我们点态结果的一个特例表明，希尔伯特空间中的每个加倍子集都具有这种意义上的点态切向量场，其维度由该集合的纳加塔（或阿苏德）维数限定。 其次，受琼斯的分析学家旅行商定理的启发，我们为希尔伯特空间中的子集引入了新的“粗略”切向量场概念，这些概念考虑了大尺度和小尺度结构。 我们证明了希尔伯特空间中的加倍子集也具有这样的粗略切向量场，同样其维度由该集合的纳加塔（或阿苏德）维数限定。 对于平面中的空洞集，这个结果可以看作是阿尔贝蒂-切尔尼亚伊-普莱斯结果的定量版本，尽管我们的结果适用于所有（甚至无限）维度。 显示英文摘要

摘要： Alberti, Cs\"ornyei and Preiss introduced a notion of a "pointwise (weak) tangent field" for a subset of Euclidean space -- a field that contains almost every tangent line of every curve passing through the set -- and showed that all area-zero sets in the plane admit one-dimensional tangent fields. We extend their results in two distinct directions. First, a special case of our pointwise result shows that each doubling subset of Hilbert space admits a pointwise tangent field in this sense, with dimension bounded by the Nagata (or Assouad) dimension of the set. Second, inspired by the Analyst's Traveling Salesman Theorem of Jones, we introduce new, "coarse" notions of tangent field for subsets of Hilbert space, which take into account both large and small scale structure. We show that doubling subsets of Hilbert space admit such coarse tangent fields, again with dimension bounded by the Nagata (or Assouad) dimension of the set. For porous sets in the plane, this result can be viewed as a quantitative version of the Alberti--Cs\"ornyei--Preiss result, though our results hold in all (even infinite) dimensions.