Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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NUMBER THEORETICAL PROPERTIES OF ROMIK'S DYNAMICAL SYSTEM

  • Cha, Byungchul (Department of Mathematics and Computer Science Muhlenberg College) ;
  • Kim, Dong Han (Department of Mathematics Education Dongguk University)
  • Received : 2019.02.10
  • Accepted : 2019.05.30
  • Published : 2020.01.31
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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

Acknowledgement

Supported by : National Research Foundation of Korea

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

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