Some constructions for the higher-dimensional three-distance theorem

Valérie Berthé¹ ; Dong Han Kim²

¹ IRIF, CNRS UMR 8243 Université Paris Diderot – Paris 7 Case 7014 75205 Paris Cedex 13, France
² Department of Mathematics Education Dongguk University – Seoul 30 Pildong-ro 1-gil, Jung-gu Seoul, 04620 Korea

Acta Arithmetica, Tome 184 (2018) no. 4, pp. 385-411

Résumé

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.

Détail

DOI : 10.4064/aa171021-30-5

Keywords: given real number alpha place fractional parts points alpha alpha dots n alpha unit circle these points partition unit circle intervals having three lengths being sum other three distance theorem consider two dimensional version three distance theorem obtained placing unit circle points alpha beta leq provide examples pairs real numbers alpha beta alpha beta rationally independent which there finitely many lengths between successive points seven lengths alpha beta badly approximable examples which there infinitely many lengths