Perturbations of measures and sets having curves in d directions

TL;DR

The paper demonstrates that sets with a weak tangent field in a metric space are typically mapped into low-dimensional sets by Lipschitz functions, with results sharp in Euclidean and certain Banach spaces.

Contribution

It establishes a link between tangent fields and the typical behavior of Lipschitz maps on measure-supported sets, extending geometric measure theory.

Findings

Typical Lipschitz maps send most of the set into low Hausdorff dimension sets.

Results are sharp in Euclidean and strictly convex Banach spaces.

Unrectifiable sets are mapped into zero-dimensional sets under typical Lipschitz maps.

Abstract

We show that whenever a separable subset $S$ of a complete metric space $X$ admits a $d$-dimensional weak tangent field, the set $S$ is close to being $d$-dimensional in the following sense. Whenever $\mu$ is a Borel finite measure on $X$ supported on $S$, then a typical $1$-Lispchitz map (in the sense of Baire category) into a Euclidean space maps $\mu$-almost all of $S$ into a set of Hausdorff dimension at most $d$. When taking $d=0$, this implies that any $1$-purely unrectifiable set is typically carried into a Hausdorff $0$-dimensional set up to a $\mu$-null set. We show that the result is sharp in Euclidean spaces and, more generally, in strictly convex Banach spaces of finite dimension.