Continued fraction expansions for complex numbers—a general approach

S. G. Dani¹

¹ Department of Mathematics Indian Institute of Technology Bombay Powai, Mumbai 400076, India

Acta Arithmetica, Tome 171 (2015) no. 4, pp. 355-369

Résumé

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.

Détail

DOI : 10.4064/aa171-4-4

Keywords: introduce general framework studying continued fraction expansions complex numbers establish results convergence corresponding sequence convergents continued fraction expansions partial quotients discrete subring mathbb analogue classical lagrange theorem characterising quadratic surds numbers eventually periodic continued fraction expansions proved monotonicity exponential growth established absolute values denominators convergents class continued fraction algorithms partial quotients ring eisenstein integers