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

$$\lim_{T\to\infty}\frac{1}{T}N_T(x,\theta)=\beta_n(x,\theta).$$

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

- Has anyone computed
*explicitly*the function $\beta_n(x,\theta)$? - 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 $n\to\infty$?

Thanks.