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Continued fractions for complex numbers and values of binary quadratic forms


Authors: S. G. Dani and Arnaldo Nogueira
Journal: Trans. Amer. Math. Soc. 366 (2014), 3553-3583
MSC (2010): Primary 11A55, 11H55; Secondary 22Fxx
DOI: https://doi.org/10.1090/S0002-9947-2014-06003-0
Published electronically: March 4, 2014
MathSciNet review: 3192607
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Abstract: We describe various properties of continued fraction expansions of complex numbers in terms of Gaussian integers. Such numerous distinct expansions are possible for a complex number. They can be arrived at through various algorithms, as also in a more general way than what we call ``iteration sequences''. We consider in this broader context the analogues of the Lagrange theorem characterizing quadratic surds, the growth properties of the denominators of the convergents, and the overall relation between sequences satisfying certain conditions, in terms of non-occurrence of certain finite blocks, and the sequences involved in continued fraction expansions. The results are also applied to describe a class of binary quadratic forms with complex coefficients whose values over the set of pairs of Gaussian integers form a dense set of complex numbers.


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Additional Information

S. G. Dani
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India
Address at time of publication: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India
Email: dani@math.tifr.res.in

Arnaldo Nogueira
Affiliation: Aix-Marseille Université, Institut de Mathématiques de Luminy, 163, avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France
Email: arnaldo.nogueira@univ-amu.fr

DOI: https://doi.org/10.1090/S0002-9947-2014-06003-0
Received by editor(s): March 4, 2011
Received by editor(s) in revised form: August 8, 2012
Published electronically: March 4, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.