Hausdorff Dimension of Union of Lines Covering a Curve: Applications to Mathematical Physics

TL;DR

This paper establishes lower bounds on the Hausdorff dimension of unions of lines covering curves and applies these results to problems in mathematical physics, revealing sharp dimension bounds in various physical contexts.

Contribution

It provides new geometric bounds on the Hausdorff dimension of line unions covering curves and applies these to conservation laws and vector fields in physics.

Findings

Union of lines covering a $C^{1,eta}$ curve has Hausdorff dimension at least $1+eta$

Spacetime observability sets for certain conservation laws have dimension at least $eta$

Line fields from vector fields with positive flux have Hausdorff dimension 3

Abstract

We prove that for any nonlinear $f \in C^{1,\alpha}([0,1])$, the union of lines covering its graph has a Hausdorff dimension of at least $1+\alpha$, and this dimension bound is sharp. We then apply these geometric results to mathematical physics, proving that spacetime observability sets for conservation laws with $\alpha$-H\"older initial wave speeds possess a dimension of at least $\alpha$. Finally, we prove that if an absolutely integrable vector field $v$ on the boundary of a polyhedron exhibits a strictly positive total flux, then the union of the line field spanned by $v$ possesses a Hausdorff dimension of 3.