Density of wave fronts

TL;DR

This work proves that wave fronts emanating from a point on a flat torus become dense, and it derives corollaries for a square billiard, the flat Klein bottle, and the cube surface. The authors develop a concise, constructive approach using elementary calculus and covering arguments: density on the flat torus is established by analyzing a small rectangular region in the universal cover and projecting to the torus, with extensions to higher-dimensional tori and to the Klein bottle. They further explore density phenomena in billiards and polyhedra, notably proving density on the cube surface via a 2:1 torus cover and projecting back, and discuss connections to geometric number theory, illumination problems, and translation surfaces. The work blends analytic, geometric, and computational perspectives to argue that density is a widespread feature even in integrable or simple settings, while outlining several open questions and directions for further study.

Abstract

We prove that wave fronts on a flat torus become dense. As a corollary, wave fronts become dense for a square billiard or for the geodesic flow on the flat Klein bottle or the cube surface.

Figures (10)

• Figure 1: What happens with a light wave front in a room of mirrors, a tsunami on a perfect sphere or a cube, and the wave front produced by a duck in a circular pond?
• Figure 2: In a Riemannian d-manifold $M$, the wave front $W_t(P)$ of a point $P$ for which a tangent space $T_PM$ exists, is the image $\exp_P(S_t)$ of the geodesic sphere $S_t=\{ |x|=t\} \subset T_PM \sim \mathbb{R}^d$ of radius $t$.
• Figure 3: To prove the result, we project a small part $R=[a,b] \times [f(a),f(b)]$ to the torus. This restriction to a smaller interval at first seems to make the problem harder. But it allows to isolate part $W_t \cap R$ of the wave front $W_t$ for which $W_t$ is a graph that is asymptotically linear for $t \to \infty$. It is then easier to see that $(W_t \cap R)/\mathbb{Z}^2$ becomes dense.
• Figure 4: Wave fronts $W_t$ on a flat torus. We prove that they become dense.
• Figure 5: Wave fronts $W_t$ on a flat Klein bottle. We prove that they become dense.

Theorems & Definitions (13)

• Definition 1: Dense Wave Front
• Conjecture 1
• Theorem 1
• proof
• Theorem 2
• Conjecture 2
• Conjecture 3
• Conjecture 4
• Conjecture 5
• Corollary 1