The number of unit distances is almost linear for most norms

Abstrakt

We prove that there exists a norm in the plane under which no n-point set determines more than O(n log n loglog 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).

Klíčová slova

• Discrete geometry
• Erdős unit distance problem
• Baire category