Billiards and the Five Distance Theorem II

Abstract

We consider a billiard table rectangle. If a billiard ball is sent out from position F(1) at the angle of $\pi/4$, then the ball will rebound against the sides of the rectangle consecutively in points $F(2),F(3),...$. Let $n\geq5$ and $\Phi= \{F(j): 1\leq j\leq n \}$ be the set of different points. An open connected subset of the perimeter of the billiard rectangle with different endpoints from the set $\Phi$ is called \textit{segment}. \textit{Length} of a segment is a distance along the perimeter between its endpoints. A segment with endpoints $F(k)$, F(l), $1\le k,l\le n$, is called \textit{even} (or \textit{odd}), and has \textit{weight} $|k-l|$ (or $k+l$) if $k$, $l$ are of the same (or different) parity. A segment is called \textit{elementary} if there are no points of the set $\Phi$ between its endpoints. Suppose $\emptyset \neq V\subseteq\{F(1),F(n)\}$. A segment $I$ is \textit{associated} with $V$ if $I$ is an elementary segment incident with an element of $V$ or $I\cap\Phi$ is nonempty set contained in $V$. Let $\omega_1<\omega_2$ be odd weights and $\omega_0$ be an even weight of segments associated with $\{F(1)\}$, and let $\omega_3<\omega_4$ be other odd weights of segments associated with $\{F(1),F(n)\}$. Suppose that $a_i$ is the length of the segment with the weight $\omega_i$, $i = 0,..., 4$. In an earlier paper the author have proved that the weights of elementary segments have at most five different values $\omega_0,..., \omega_4$. Moreover, elementary segments with equal weights have equal lengths. Let $A_i$ be the set of all elementary segments with weight $\omega_i$. In this paper we prove that, if we know weights $\omega_0$, $\omega_1$, $\omega_4$, and $\epsilon, \delta \in \{-1, 1\}$ such that $a_2-\epsilon a_1 = a_3-\delta a_4 =a_0$, then we can easily calculate $|A_0|, ..., |A_4|$.