The three gap theorem and the space of lattices

TL;DR

This paper offers a novel approach to the three gap theorem by utilizing the space of two-dimensional Euclidean lattices, providing new insights into the distribution of fractional parts of scaled sequences.

Contribution

It introduces a new proof of the three gap theorem leveraging lattice space techniques, differing from previous methods.

Findings

New proof of the three gap theorem using lattice space

Enhanced understanding of fractional part distributions

Potential applications to related problems in number theory

Abstract

The three gap theorem (or Steinhaus conjecture) asserts that there are at most three distinct gap lengths in the fractional parts of the sequence $\alpha,2\alpha,\ldots,N\alpha$, for any integer $N$ and real number $\alpha$. This statement was proved in the 1950s independently by various authors. Here we present a different approach using the space of two-dimensional Euclidean lattices.