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## versão impressa ISSN 0034-7426

### Rev.colomb.mat. vol.54 no.1 Bogotá jan./jun. 2020

#### https://doi.org/10.15446/recolma.v54n1.89777

Original articles

Deducing Three Gap Theorem from Rauzy-Veech induction

Deduciendo el teorema de las tres brechas vía inducción Rauzy-Veech

1Ruhr West University of Applied Sciences, Mülheim a. d. Ruhr, Germany

Abstract:

The Three Gap Theorem states that there are at most three distinct lengths of gaps if one places n points on a circle, at angles of z, 2z, … nz from the starting point. The theorem was first proven in 1958 by Sós and many proofs have been found since then. In this note we show how the Three Gap Theorem can easily be deduced by using Rauzy-Veech induction.

Keywords: Three Gap Theorem; Rauzy-Veech induction; Kronecker sequence; interval exchange transformation; uniform distribution

Resumen:

El teorema de las tres brechas indica que existen a lo sumo tres longitudes distintas de brechas si se sitúan n puntos en un círculo, en ángulos z, 2z, … nz a partir del punto inicial. El teorema se demostró primero en 1958 por Sós y muchas pruebas han sido encontradas desde entonces. En esta nota mostramos cómo el teorema de las tres brechas puede ser fácilmente deducido usando inducción de tipo Rauzy-Veech.

Palabras clave: Teorema de las tres brechas; inducción Rauzy-Veech; sucesión de Kronecker; intercambio de intervalos; distribución uniforme

Full text available only in PDF format.

Acknowledgment.

The author thanks the anonymous referees for their very valuable comments.

REFERENCES

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Received: March 06, 2019; Accepted: February 11, 2020

*Correspondencia: Christian Weiss, Institute of Natural Sciences, Ruhr West University of Applied Sciences, Duisburger Str. 100, D-45479 Mülheim an der Ruhr, Germany. Correo electrónico: christian.weiss@hs-ruhrwest.de. DOI: https://doi.org/10.15446/recolma.v54n1.89777

2010 Mathematics Subject Classification. 11J71, 11K31, 37A10