Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2020; 57(1): 251-274

Online first article July 23, 2019      Printed January 31, 2020

https://doi.org/10.4134/BKMS.b190163

Copyright © The Korean Mathematical Society.

Number theoretical properties of Romik's dynamical system

Byungchul Cha, Dong Han Kim

Muhlenberg College; Dongguk University

Abstract

We study a dynamical system that was originally defined by Romik in 2008 using an old theorem of Berggren concerning Pythagorean triples. Romik's system is closely related to the Farey map on the unit interval which generates an additive continued fraction algorithm. We explore some number theoretical properties of the Romik system. In particular, we prove an analogue of Lagrange's theorem in the case of the Romik system on the unit quarter circle, which states that a point possesses an eventually periodic digit expansion if and only if the point is defined over a real quadratic extension field of rationals.

Keywords: Pythagorean triple, continued fraction, Berggren theorem, Romik system

MSC numbers: Primary 11J70; Secondary 11A55

Supported by: Research supported by the National Research Foundation of Korea (NRF-2018R1A2B 6001624).