Billiard and the five-gap theorem

Abstract

Let us consider the interval [ 0 , 1 ) as a billiard table rectangle with perimeter 1 and a sequence 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 I is an open segment of a billiard rectangle, then the differences between the successive values of m for which the F ( m ) lies in I , take at most one even and at most four distinct odd values.