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arXiv · 1703.01620

Direction sets, Lipschitz graphs and density

Alex Iosevich·Jonathan Pakianathan
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Abstract

We consider the direction set determined by various subsets $E$ 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 $E$.

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Alex Iosevich, Jonathan Pakianathan. 2017-03-05. Direction sets, Lipschitz graphs and density. https://arxiv.org/abs/1703.01620

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