Direction sets, Lipschitz graphs and density
Classical Analysis and ODEs
2017-03-07 v1 Combinatorics
Abstract
We consider the direction set determined by various subsets of Euclidean space and show that there is a trichotomy: Either (i) The subset is the graph of a Lipschitz function and the direction set is not dense in the sphere, (ii) The subset is the graph of a non-Lipschitz function and the direction set is dense but not everything, or (iii) The subset is not a graph (in a suitable sense) and every direction is determined by the set. We then explore a variety of results based on this trichotomy under additional assumptions on the set .
Cite
@article{arxiv.1703.01620,
title = {Direction sets, Lipschitz graphs and density},
author = {Alex Iosevich and Jonathan Pakianathan},
journal= {arXiv preprint arXiv:1703.01620},
year = {2017}
}