Marstrand's density theorem for arbitrary norms in the plane

TL;DR

This paper proves that in the plane, measures with positive finite density at almost every point with respect to any norm must have an integer dimension, extending classical results to arbitrary norms.

Contribution

It generalizes Marstrand's density theorem to arbitrary norms in the plane, establishing that the dimension must be an integer under these conditions.

Findings

Dimension must be an integer for measures with positive finite density in arbitrary norms.

Extends classical density theorems to more general geometric settings.

Provides new insights into measure theory and geometric analysis in the plane.

Abstract

We show that if a non-trivial measure in the plane admits, at almost every point, positive and finite $\alpha$-dimensional density with respect to some norm, then $\alpha$ must be an integer.