Badly approximable numbers over imaginary quadratic fields

Abstract

We recall the notion of nearest integer continued fractions over the Euclidean imaginary quadratic fields K and characterize the "badly approximable" numbers, (z such that there is a C = C(z) > 0 with |z − p/q| ≥ C/|q| 2 for all p/q ∈ K), by boundedness of the partial quotients in the continued fraction expansion of z. Applying this algorithm to "tagged" indefinite integral binary Hermitian forms demonstrates the existence of entire circles in C whose points are badly approximable over K, with effective constants.By other methods (the Dani correspondence), we prove the existence of circles of badly approximable numbers over any imaginary quadratic field. Among these badly approximable numbers are algebraic numbers of every even degree over Q, which we characterize. All of the examples we consider are associated with cocompact Fuchsian subgroups of the Bianchi groups SL 2 (O), where O is the ring of integers in an imaginary quadratic field.

Generalized continued fraction expansions of complex numbers, and applications to quadratic and badly approximable numbers

“…Since |a n | ≤ |z n | + 1 for all n this proves the last statement in the Corollary. Corollary 3.7 generalizes a result of Hines [5] proved in the special case when Γ is the ring of Gaussian integers, the continued fraction expansion is with respect to the nearest integer algorithm, and P has no root in K. The partial quotients in the continued fraction expansion of a z ∈ C ′ being bounded relates to z being "badly approximable" by elements of K. This connection was discussed in [5] in the case of the expansions with respect to the nearest integer algorithm, for the ring of Gaussian integers. In the next section we discuss the approximability issues in the broader context of the more general continued fraction expansions as considered in the earlier sections here, and relate the property of the partial quotients being bounded to bad approximability, under certain general conditions on the expansions.…”

“…We next apply Theorem 3.1 and prove the following analogue for hermitian quadratic polynomials, viz. functions of the form H(z, 1), where H is a Hermitian binary form, showing that if z ∈ C ′ is a root of H(z, 1) then, under certain general conditions, the partial quotients {a n } are bounded; this extends a result of Hines [5], where it is proved in the special case of the nearest integer algorithm. We begin by noting the following.…”

“…No strict equivalent of the above is known for any algorithmic expansions of complex numbers (see however [9] for exploration on this question in another direction). There are however weaker versions of the result, proving the analogous relation upto a constant multiple (see [5] and other references cited there). In this respect it turns out to be more fruitful to compare |qz − p| with |q n z − p n |, where p, q ∈ Γ (the latter being a Euclidean subring of C as before), q = 0 and an appropriate convergent of z, in place of |z − p q | and |z − pn qn |; we note that in the setting as above, if |qz − p| ≥ c|q n z − p n |, for some c > 0, then…”

“…In another direction, R. Hines [5] noted, more recently, the existence of circles in C all of whose points are badly approximable with respect to Gaussian integers, via a study of the continued fraction expansion of zeroes of Hermitian forms, with respect to the Hurwitz algorithm.…”

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).…”