Billiard and the five-gap theorem

Let us consider the interval [0,1)[0,1) as a billiard table rectangle with perimeter 1 and a sequence F(m)∈[0,1),m∈N∪{0}F(m)∈[0,1),m∈N∪{0}, of successive rebounds of a billiard ball against the sides of a billiard rectangle. We prove that if II is an open segment of a billiard rectangle, then the differences between the successive values of mm for which the F(m)F(m) lies in II, take at most one even and at most four distinct odd values.