Some constructions for the higher-dimensional three-distance theorem

Volume 184 / 2018

Valérie Berthé, Dong Han Kim Acta Arithmetica 184 (2018), 385-411 MSC: Primary 11J13; Secondary 11J70, 11J71, 11B75, 11A55. DOI: 10.4064/aa171021-30-5 Published online: 30 July 2018


For a given real number $\alpha$, let us place the fractional parts of the points $0, \alpha, 2 \alpha, \dots, (N-1) \alpha$ on the unit circle. These points partition the unit circle into intervals having at most three lengths, one being the sum of the other two. This is the three-distance theorem. We consider a two-dimensional version of the three-distance theorem obtained by placing on the unit circle the points $ n\alpha+ m\beta $ for $0 \leq n,m \lt N$. We provide examples of pairs of real numbers $(\alpha,\beta)$, with $1,\alpha, \beta$ rationally independent, for which there are finitely many lengths between successive points (and in fact, seven lengths), with $(\alpha,\beta)$ not badly approximable, as well as examples for which there are infinitely many lengths.


  • Valérie BerthéIRIF, CNRS UMR 8243
    Université Paris Diderot – Paris 7
    Case 7014
    75205 Paris Cedex 13, France
  • Dong Han KimDepartment of Mathematics Education
    Dongguk University – Seoul
    30 Pildong-ro 1-gil, Jung-gu
    Seoul, 04620 Korea

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