Continued fraction expansions for complex numbers—a general approach

Volume 171 / 2015

S. G. Dani Acta Arithmetica 171 (2015), 355-369 MSC: Primary 11J70; Secondary 11J25. DOI: 10.4064/aa171-4-4


We introduce a general framework for studying continued fraction expansions for complex numbers, and establish some results on the convergence of the corresponding sequence of convergents. For continued fraction expansions with partial quotients in a discrete subring of $\mathbb C$ an analogue of the classical Lagrange theorem, characterising quadratic surds as numbers with eventually periodic continued fraction expansions, is proved. Monotonicity and exponential growth are established for the absolute values of the denominators of the convergents for a class of continued fraction algorithms with partial quotients in the ring of Eisenstein integers.


  • S. G. DaniDepartment of Mathematics
    Indian Institute of Technology Bombay
    Powai, Mumbai 400076, India

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