This is a question in a rather well investigated subject of which I know very little and I have a hard time "translating" the general results available. Let me also say that I got interested in this question after a conversation with a friend about placements of cell phone towers. (It is too long a story to include here.)

Here's the problem. $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\bsP}{\boldsymbol{P}}$ $\newcommand{\bR}{\mathbb{R}}$ Fix a positive integer $n$ and let $\bsP_n$ denote the regular $n$-gon centered at the origin and inscribed in the unit circle.

The billiard in the title refers to the motion of a unit speed photon inside $\bsP_n$ assuming the edges of $\bsP_n$ are perfect mirrors. For every $x$ in the interior of $\bsP_n$, any unit vector identified with a $\theta\in S^1$ and any $T>0$ we denote by $N_T(x,\theta)$ the number of "bounces-off-mirrors" during the time interval $[0,T]$ of a photon that starts its motion at $x$ in the direction $\theta$. The question involves the behavior of $N_T(x,\theta)$ for large $T$.

I believe that there exists a measurable function $\beta_n:\bsP_n\times S^1\to [0,\infty)$ such that for almost all $(x,\theta)\in\bsP_n\times S^1$ we have


The questions involve the proportionality constants $\beta_n(x,\theta)$.

  1. Has anyone computed explicitly the function $\beta_n(x,\theta)$?
  2. Has anyone computed explicitly the average

$$\bar{\beta}_n(x):=\frac{1}{2\pi} \int_0^{2\pi}\beta_n(x,\theta) d\theta ?$$

(I am inclined to believe that $\bar{\beta}_n(x)$ is constant.)

Above, the cases $n=3,6$ are of particular interest in the original cell phone tower problem.

Finally, has anyone investigated the behavior of $\beta_n(x,\theta)$ as $n\to\infty$?



The answer is given by the general theory of rational billiard flows (i.e., those on polygons whose angles are rational multiples of $\pi$). On such a polygon $Q$ the tangent vectors to any given orbit are parallel to a finite set of unit vectors, so that the orbits with initial direction $\theta$ lie on an invariant surface $M_\theta$ which consists of a finite number of copies of $Q$, one for each potential direction of an orbit with initial direction $\theta$. According to a theorem of Kerckhoff, Masur and Smillie (http://www.ams.org/mathscinet-getitem?mr=855297 or the announcement http://www.ams.org/mathscinet-getitem?mr=799797), for almost all $\theta$ the flow on $M_\theta$ is uniquely ergodic. It implies that for a.e. $\theta$ the projection to $Q$ of every orbit with initial direction $\theta$ is uniformly distributed in $Q$ (with respect to the Lebesgue measure).

Therefore, $1/\beta_n(x,\theta)$ coincides, for a.e. $\theta$ and any $x\in Q$, with the average length of segments in billiard orbits, i.e., by the ergodic theorem, with the average of the function $L(x,\theta)$ (the distance between the points where the line issued from $x$ in the direction $\theta$ intersects the boundary of $Q$) with respect to the Lebesgue measure on $M_\theta$, which only depends on $\theta$ and should be quite easy to caclulate explicitly in the case of regular polygons.


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