## Continued fraction expansions for complex numbers—a general approach

### Volume 171 / 2015

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

#### Abstract

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.