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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 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. Dani. "Continued fraction expansions for complex numbers-a general approach." Acta Arithmetica 171.4 (2015): 355-369. <http://eudml.org/doc/279347>.
@article{S2015, 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 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.}, author = {S. G. Dani}, journal = {Acta Arithmetica}, keywords = {continued fraction expansions of complex numbers; algorithms; Eisenstein integers; Lagrange theorem; growth of denominators of convergents}, language = {eng}, number = {4}, pages = {355-369}, title = {Continued fraction expansions for complex numbers-a general approach}, url = {http://eudml.org/doc/279347}, volume = {171}, year = {2015}, }
TY - JOUR AU - S. G. Dani TI - Continued fraction expansions for complex numbers-a general approach JO - Acta Arithmetica PY - 2015 VL - 171 IS - 4 SP - 355 EP - 369 AB - 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 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. LA - eng KW - continued fraction expansions of complex numbers; algorithms; Eisenstein integers; Lagrange theorem; growth of denominators of convergents UR - http://eudml.org/doc/279347 ER -