The three gap theorem and the space of lattices

Jens Marklof; Andreas Strömbergsson

The three gap theorem and the space of lattices

Number Theory 2017-06-23 v2

Authors: Jens Marklof , Andreas Strömbergsson

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.

Keywords

three gap theorem Steinhaus conjecture fractional parts Euclidean lattices gap lengths number theory sequence analysis

Cite

@article{arxiv.1612.04906,
  title  = {The three gap theorem and the space of lattices},
  author = {Jens Marklof and Andreas Strömbergsson},
  journal= {arXiv preprint arXiv:1612.04906},
  year   = {2017}
}

Comments

To appear in the American Mathematical Monthly

The three gap theorem and the space of lattices

Abstract

Keywords

Cite

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