Pseudotangents to Lipschitz curves

TL;DR

This paper explores the structure of pseudotangents to Lipschitz curves, extending previous results to include a broader class of sets, and discusses open questions about the universality of pseudotangents.

Contribution

It generalizes the understanding of pseudotangents to Lipschitz curves by characterizing sets where these pseudotangents can be obtained, and raises open problems about their universality.

Findings

Sets where pseudotangents are obtained can be any compact, uniformly disconnected set with Lipschitz capture.

The paper extends previous results to a broader class of sets.

Open question remains whether Lipschitz curves can obtain all pseudotangents at every point.

Abstract

In this paper, we extend the result of arXiv:2409.13662 by showing that the set on which every pseudotangent is obtained on a Lipschitz curve can be any compact, uniformly disconnected set in Euclidean space which admits any Lipschitz capture. We do not obtain a characterization of such sets however, indeed we leave open the very strong question of whether or not a Lipschitz curve can obtain every pseudotangent at every point.