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Revista Colombiana de Matemáticas
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
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
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
Acknowledgment.
The author thanks the anonymous referees for their very valuable comments.
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Received: March 06, 2019; Accepted: February 11, 2020