Complex Numbers With Bounded Partial quotients

Abstract

Abstract. Conjecturally, the only real algebraic numbers with bounded partial quotients in their regular continued fraction expansion are rationals and quadratic irrationals. We show that the corresponding statement is not true for complex algebraic numbers in a very strong sense, by constructing for every even degree d algebraic numbers of degree d that have bounded complex partial quotients in their Hurwitz continued fraction expansion. The Hurwitz expansion is the generalization of the nearest integer continued fraction for complex numbers. In the case of real numbers the boundedness of regular and nearest integer partial quotients is equivalent. IntroductionReal numbers admit regular continued fraction expansions that are unique except for an ambiguity in the ultimate partial fraction of rational numbers. The same is true for nearest integer continued fraction expansions. The nearest integer expansion is easily obtained from the regular expansion, by applying a certain modification for partial quotients that equal 1, as a result of which some partial quotients are incremented by 1, and some minus signs are introduced. As a consequence, for questions concerning the boundedness of partial quotients of real numbers, the behaviour of regular and nearest integer continued fractions is alike.In both cases, finite expansions occur precisely for the rational numbers, and ultimately periodic expansions occur precisely for quadratic irrational numbers. Not much is known about the partial quotients for other algebraic irrationalities. There exist transcendental numbers with bounded partial quotients, and also transcendental numbers with unbounded partial quotients.Date: September 12, 2011.

From Binary Hermitian Forms to parabolic cocycles of Euclidean Bianchi groups

“…To prove the continuity of H 1,∆ we use Bengoechea's methods but the difficulty with that approach is finding a continued fraction algorithm for complex numbers that is similar to the classical continued fraction (i.e., nearest integer continued fraction) expansion of real numbers which Bengoechea uses in her paper. Such an algorithm was developed by Hurwitz [16] and has been studied recently by a number of different people (see [2,7,8,9,15], and other references cited therein).…”

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.…”