A dichotomy of sets via typical differentiability

TL;DR

This paper characterizes when an analytic set in Euclidean space contains points where a typical Lipschitz function is differentiable, revealing a dichotomy based on the set's geometric covering properties.

Contribution

It provides a new criterion linking the geometric structure of sets to the differentiability points of typical Lipschitz functions, highlighting a sharp dichotomy.

Findings

Sets not coverable by purely unrectifiable sets contain differentiability points.

Coverable sets by such unrectifiable sets have no differentiability points for typical functions.

A dichotomy in the differentiability behavior of Lipschitz functions based on set structure.

Abstract

We obtain a criterion for an analytic subset of a Euclidean space to contain points of differentiability of a typical Lipschitz function, namely, that it cannot be covered by countably many sets, each of which is closed and purely unrectifiable (has zero length intersection with every $C^1$ curve). Surprisingly, we establish that any set failing this criterion witnesses the opposite extreme of typical behaviour: In any such coverable set a typical Lipschitz function is everywhere severely non-differentiable.