Coarse and pointwise tangent fields

Abstract

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.