The number of unit distances is almost linear for most norms

TL;DR

This paper demonstrates that for most norms in the plane, the maximum number of unit distances determined by n points grows almost linearly, specifically bounded by O(n log n log log n).

Contribution

It shows that most norms have at most nearly linear unit distance counts, establishing a generic property in the space of all planar norms.

Findings

Most norms have at most O(n log n log log n) unit distances for n-point sets.

The set of norms with this property is comeager in the space of all norms.

The result applies to a generic class of norms in the plane.

Abstract

We prove that there exists a norm in the plane under which no n-point set determines more than O(n log n log log n) unit distances. Actually, most norms have this property, in the sense that their complement is a meager set in the metric space of all norms (with the metric given by the Hausdorff distance of the unit balls).