Continued fractions for complex numbers and values of binary quadratic forms

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.

Finite Partitions for Several Complex Continued Fraction Algorithms

“…an assuming a n = 0 and q n = 0; see [6,Theorem 2.2] for a sufficient condition to imply q n = 0 ∀ n. When it exists, the term p n /q n is called the n th convergent of the continued fraction.…”

“…The following combines terminology from Dani-Nogueira [6] and notation from Katok-Ugarcovici [14,15]. 1 Definition 2.1.…”

“…After then defining {p n } and {q n } by (1), the sequence { pn qn } will either terminate (in which case the final term pn qn = z) or will converge to z. Terminating sequences only occur for z ∈ Q[ ], see [6], so we will focus only on z ∈ C \ Q[ ].…”

“…Continued fractions for complex numbers have been studied from a number theoretic perspective by Adolf Hurwitz [9], Doug Hensley [8], and more recently by S. G. Dani and Arnaldo Nogueira [6].…”

Calculations of the Invariant Measure for Hurwitz Continued Fractions

“…The simplest complex CF expansion is the Hurwitz complex CF (see Section 2 for definitions). 1 While the Hurwitz complex CF is well-studied [1,4,5,7,10], little is yet known about the corresponding invariant measure. It is known (see Theorem 2.1) that the density h of the invariant measure is piece-wise real-analytic with 12 pieces of analyticity and it is known that it satisfies certain symmetries, but that is all.…”