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

Hanwen Liu

Abstract

We prove that for any nonlinear $f \in C^{1,α}([0,1])$, the union of lines covering its graph over a sufficiently large full measure subset has a Hausdorff dimension of at least $1+α$, and this dimension bound is sharp. We then apply these geometric results to mathematical physics, proving that light rays forming a differentiable caustic in the plane must illuminate a 2D region, and that spacetime observability sets for conservation laws with $α$-Hölder initial wave speeds possess a dimension of at least $α$. Finally, we establish a measure theoretic result: For a continuous differentiable function $f$ of which derivative is non-constant of bounded variation, if the union of some family of lines that cover $\operatorname{graph}(f)$ has Hausdorff dimension less than $2$, then the distributional derivative $f''$ is a singular measure.

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

Abstract

We prove that for any nonlinear , the union of lines covering its graph over a sufficiently large full measure subset has a Hausdorff dimension of at least , and this dimension bound is sharp. We then apply these geometric results to mathematical physics, proving that light rays forming a differentiable caustic in the plane must illuminate a 2D region, and that spacetime observability sets for conservation laws with -Hölder initial wave speeds possess a dimension of at least . Finally, we establish a measure theoretic result: For a continuous differentiable function of which derivative is non-constant of bounded variation, if the union of some family of lines that cover has Hausdorff dimension less than , then the distributional derivative is a singular measure.
Paper Structure (8 sections, 21 theorems, 75 equations, 5 figures)

This paper contains 8 sections, 21 theorems, 75 equations, 5 figures.

Table of Contents

  1. Introduction and Background
  2. Graphs of Differentiable Functions
  3. Applications to Geometric Optics
  4. Hölder and Sobolev Spaces
  5. Applications to Conservation Laws
  6. Functions of Bounded Variation
  7. Methods from Topology
  8. Generalization to Hypersurfaces

Key Result

Lemma 1.1

Let $\Lambda$ be a measurable subset of $\mathbb{R}^2$, and $\pi\colon\mathbb{R}^2\rightarrow\mathbb{R}^1$ the projection onto the first coordinate. If $\dim_\mathcal{H}(\pi(\Lambda))\geq1$, then the Hausdorff dimension of is exactly equal to $2$.

Figures (5)

  • Figure 1.1: The Sierpiński triangle of Hausdorff dimension $\alpha=\log(3)/\log(2)\approx1.585$.
  • Figure 2.1: An example in $\mathbb{R}^2$: The linear function on the left of which union of tangent lines has Hausdorff dimension 1, comparing with the strictly convex function on the right of which union of tangent lines has Hausdorff dimension 2.
  • Figure 3.1: Geometric visualization of ray intersection forming a caustic. A nonlinear wavefront set forces emanating normal lines to overlap, creating a possibly singular 1D envelope.
  • Figure 4.1: The fractional Devil's staircase, which serves as the derivative $f'(x)$ in our sharp dimensional construction, demonstrating constant intervals over the gaps of the Cantor set.
  • Figure 5.1: Characteristic lines in the space-time plane for Burger's equation. A linear initial wave speed yields perfect shock focusing, while a general non-linear $C^{1,\alpha}$ wave speed creates a smeared caustic requiring a higher-dimensional sensor.

Theorems & Definitions (41)

  • Lemma 1.1: Falconer-Mattila, 2016 Falconer2016
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Theorem 3.1
  • Remark 3.2
  • Theorem 4.1: Jankov-von Neumann, 1949 Kechris1995
  • ...and 31 more