A concise geometric proof of the three distance theorem

Abstract

The three distance theorem states that for any given irrational number $\alpha$ and a natural number $n$, when the interval $( 0, 1 )$ is divided into $n+1$ subintervals by integer multiples of $\alpha$, namely, $\{0\}, \{ \alpha \}, \{ 2\,\alpha \},\dots, \{ n\,\alpha \}$, then each subinterval is limited to at most three different lengths. Steinhaus conjectured this theorem in the 1950s, and many researchers have given various proofs since then. This paper aims to improve the perspective by showing a two-dimensional map which tells how the unit interval is divided by continuously changing $\alpha$, and provide a concise proof of the theorem. By illustrating this proof through geometric visualizations, we offer a clearer and more intuitive understanding of the underlying principles and relationships. The approach not only reinforces the classical results but also paves the way for new insights and applications in the study of irrational numbers and their properties. Additionally, we present a simple proof of the three gap theorem, which is a dual of the three distance theorem.