Continued Fractions on the Heisenberg Group
Abstract.
We provide a generalization of continued fractions to the Heisenberg group. We prove an explicit estimate on the rate of convergence of the infinite continued fraction and several surprising analogs of classical formulas about continued fractions. We then discuss dynamical properties of the associated Gauss map, comparing them with base expansions on the Heisenberg group and continued fractions on the complex plane.
2010 Mathematics Subject Classification:
Primary 22E40, 11J70, Secondary 53C171. Introduction
A regular continued fraction (RCF) expansion represents an irrational number as
(1.1) 
The integers are the continued fraction digits of (also called the partial quotients). Regular continued fractions and their many variations have played an important part in Diophantine approximation, hyperbolic geometry, and the study of quadratic irrationals.
Many higherdimensional generalizations of continued fractions have been developed to extend this powerful theory, but these efforts have met with varying success. In this paper, we develop a notion of continued fractions in the noncommutative setting of the Heisenberg group (in a sense, a complex twodimensional continued fraction). Surprisingly, we recover not only standard results of convergence (see Theorem 1.3), but also several simple, direct analogs of classical formulas for regular continued fractions—formulas which lack simple analogs for any other known multidimensional continued fraction. This suggests that continued fractions are a reasonable and natural object of study on the Heisenberg group.
This paper provides the basic properties of Heisenberg continued fractions, and opens up the way for many new questions. In future papers, we intend to link our study to that of Diophantine approximation on the Heisenberg group (see [7]) and the behavior of geodesics in complex hyperbolic space (see Remark 2.9 and [16]). Additional interesting questions include extending these results to similar spaces and their lattices (specifically, we expect our results to hold for all boundaries of hyperbolic rankone symmetric spaces), a characterization of periodic continued fraction expansions, a careful analysis of the dynamical properties of the associated Gauss map, and a description of the dual space.
The setting for this paper will be the Heisenberg group , arguably the most natural noncommutative generalization of . Specifically, is with the modified group law (which we denote by )
(1.2) 
Note that in the first two coordinates one sees the usual addition of vectors, while the third coordinate incorporates an antisymmetric term. Note also that the group inverse of an element is given by .
Let denote the set of points in with all integer coordinates. These form a subgroup of , and we will think of them as the integers within . Likewise, we think of points with all rational coordinates, , as rational points.
Given a generic point there is a unique nearest Heisenberg integer , with respect to the Heisenberg group’s standard gauge metric:
(1.3) 
Note that left translations by elements of are isometries. That is, for all . In addition, one has an inversion operation (see §2.1) satisfying
Given a point , we may remove the integer part of via .
Definition 1.1.
The continued fraction digits and forward iterates of a point are defined inductively by:
Note that is undefined. Thus, the process may terminate after finitely many steps. We will characterize points for which this happens in Theorem 3.10 and, for the majority of the paper, focus our attention on points with infinitely many digits. We will also generally assume that unless otherwise specified.
Definition 1.2.
Let be a sequence of elements of . For a finite sequence, define the associated continued fraction,
supressing product notation and parentheses. It is clear that if is finite, then .
For an infinite sequence, we write
provided the limit exists.
Our main result is to show that and define a valid notion of a continued fraction expansion for a point in . Namely, we prove
Theorem 1.3.
The following properties hold:

Let be a sequence of elements of satisfying for each . Then exists and is unique regardless of whether is finite or infinite (Theorem 3.7).

A point satisfies for a finite sequence of elements of if and only if (Theorem 3.10).

Every point in has a continued fraction expansion. That is, for all , the limit is unique and equal to (Theorem 3.21).
Throughout §3, we obtain variants of classical continued fraction results. We show a relationship between the denominator of a rational point and the length of its continued fraction expansion in Theorem 3.11. We find a recursive formula for the approximants in Theorem 3.18, and show that the distance between and its approximants satisfies a variant of a classical relation in Theorem 3.23. We prove that the convergence of is uniform on a fullmeasure set in Theorem 3.24.
In §4 we consider a generalization of the classical Gauss map . Namely, let be the Dirichlet region for , defined as the set of points such that . It is easy to see that is a fundamental region for , that is, the translates of by elements of tile without overlap. It is also clear that for all , one has .
We define a function on the Dirichlet region by . We ask whether admits a invariant measure absolutely continuous with respect to Lebesgue measure, and whether is ergodic with respect to this measure (Questions 4.1 and 4.2). We demonstrate the difficulties in studying these questions by discussing continued fractions on the complex plane , and answer them positively for the simpler base expansions in .
We will now recall some background on classical continued fractions (§1.1) and the Heisenberg group (§2), and then study Heisenberg continued fractions in §3 and discuss their dynamical properties in §4.
1.1. Classical Theory of Continued Fractions
There are many variants on classical continued fractions and many ways to approach them (for good general references, see [3, 6, 8, 11]). We shall examine Nakada’s continued fractions, since the study of them bears the most immediate resemblance to the Heisenberg continued fractions we examine in this paper (see also §4.1 for a discussion of continued fractions on ). The continued fractions have two wellknown continued fraction variants as special cases: Regular Continued Fractions (when ) and Nearest Integer Continued Fractions (when ).
Let . Define the the Gauss map by
where is the unique integer such that Most continued fractions variants begin with these three simple pieces: a fundamental domain ( here), an inversion that takes a point out of the fundamental domain (), and a piecewise linear translation that shifts us back into the fundamental domain ().
The digits of the continued fraction expansion for a number consist of two parts, , where
The sequence of digits terminates when . These digits serve to record the data that is lost by iterating the noninjective map . In particular, we have
Note that , so that acts as a forward shift of the continued fraction digits of .
One of the fundamental objects of study in the field of continued fractions is the sequence of convergents or rational approximants for a number , given by
It is often easier to understand abstract properties of the sequence of convergents for a number , than it is to understand abstract properties of the whole continued fraction expansion for .
A particularly useful property of convergents is the following matrix relation:
(1.4) 
From this relation, one can derive the recurrence relation . While it would be nice to know that the form an increasing sequence of positive integers, this is not always the case (such as with continued fractions with odd partial quotients [1]).
We can treat matrices as Möbius transforms, via
If we do this, then the simple relation
together with (1.4), implies the more interesting relation
(1.5) 
By solving for (or by applying the inverse of the matrix to both sides), one can obtain
(1.6) 
Careful—but elementary—manipulation of the formulas (1.5) and (1.6) yields
(1.7) 
From (1.7) it is short exercise to see that converges to , and hence that converges to . Thus it makes sense to write as an infinite continued fraction expansion
There are varying notions of convergence for continued fractions variants besides the fact that tends to , which is typically known as weak convergence. In multidimensional continued fractions, where one might have convergents
the property that tends to for all is known as strong convergence. (The JacobiPerron continued fraction, which is in many ways considered to be the prototypical multidimensional continued fraction, does not satisfy strong convergence.) The fact that all columns of the matrices (1.4) converge (projectively) to the same point is known as uniform convergence. Uniform convergence is nontrivial for higherdimensional continued fraction variants.
In general, it is hard to know whether an arbitrary sequence of continued fraction digits produces a convergent infinite continued fraction. (Even the seemingly innocuous twodigit sequence causes convergence problems.) One major result on this question is Pringsheim’s theorem, which states that if for all , then the infinite continued fraction converges. For more on this topic, see [18].
For many continued fractions, the digit shift map is ergodic with respect to some invariant measure. For Regular Continued Fractions (), the invariant measure that is absolutely continuous with respect to Lebesgue is the classic Gauss measure
The ergodicity of the map means that there is a notion of average behavior for continued fractions. For example, for almost all , the regular continued fraction expansion of satisfies
and approximately percent of the time.
Applications of continued fractions come from various areas. We mention only a few in greater detail here. One of the most classical results on continued fractions is Lagrange’s Theorem, which states that has an eventually periodic continued fraction expansion if and only if is a quadratic irrational number: thus one often studies properties of quadratic irrationals by understanding their RCF expansion. The term that appeared in (1.7) is closely related to the study of best approximants—namely, rational numbers that satsify the following relation
must be an RCF convergent for .
2. The Heisenberg Group
We will think of the Heisenberg group in three different ways. For geometric purposes, including illustration and discussion of measures, we will identify with (with the appropriate group structure and geometry). For the majority of the paper, however, we will be concerned with the representation of as a group of unitary matrices or as a subset of . This is in direct analogy with thinking of the real numbers as elements of or as the real axis within . We now discuss these models, and then record some information on discrete subgroups of and their fundamental domains.
We emphasize that the topological and measuretheoretic notions we consider do not (qualitatively) depend on the model we choose, nor on the metric. In particular, convergence in can be shown using the intrinsic gauge metric, or using metrics intrinsic to the model, such as the Euclidean metrics on or .
2.1. Geometric Model
In the introduction, we defined as the space with group law
Combining the first two coordinates into a complex number, becomes with group law
We will think of these as the same model, and use it primarily when geometry or visualization are concerned. There are several standard (topologically equivalent) metrics on ; we will work with the gauge metric. The gauge and distance are defined by:
There are four basic transformations we are interested in:

Left translations , for ,

Rotations , for ,

Metric dilations , for ,

The Koranyi inversion given by
Translations and rotations do not distort distances or volume (that is, the Lebesgue measure on ). The map is a group homomorphism dilating distances by a factor of and volume by a factor of . The Koranyi inversion is a conformal map with the following important property.
Lemma 2.1 (See p.19 of [2]).
Let . One has
In particular, one has , so that the inside and outside of the unit ball are interchanged. Note that individual points on the unit sphere are not fixed.
Remark 2.2.
We record the following relationship between volumes and radii of balls in . In particular, the lemma implies that the Heisenberg group has Hausdorff dimension 4, and that the Lebesgue measure is equivalent to the Hausdorff measure on .
Lemma 2.3.
The volume of a ball of radius around a point is given by
Proof.
Applying a left translation, we may assume is the origin. Further, we may rescale by the Heisenberg dilation to obtain . The dilation distorts by a factor of . ∎
Lemma 2.4.
The Jacobian determinant of the Koranyi inversion at a point is given by .
2.2. Real Nilpotent Model
It is common to describe as the group of nilpotent uppertriangular 3by3 real matrices. Our definition is related to this real nilpotent model via the Lie group isomorphism
(2.1) 
We will not use the real nilpotent model, although our results can be rephrased for it. Note that under (2.1), is not identified with matrices with integer entries.
2.3. Unitary Representation
For calculation purposes, we will use the (Siegel) unitary representation of . Namely, we will embed in via the homomorphism:
(2.2) 
Remark 2.5.
In literature, one sees a factor of rather than in the embedding. The latter is more convenient for our purposes.
Let be the Hermitian inner product given by
In particular, we record
(2.3) 
We will refer to a vector of norm as a null vector.
Abusing notation, we will also use to denote the skewdiagonal matrix above. Note that has signature : it has two positive and one negative eigenvalue.
The unitary group is the set of matrices satisfying for all . Equivalently, satisfies , where denotes the conjugate transpose. We will additionally distinguish the subgroups and consisting of matrices satisfying, respectively, or . We have:
Lemma 2.6.
.
Later calculations will require us to step outside of . The following lemma provides a basic property of elements of unitary matrices.
Lemma 2.7.
Every matrix in is of the form
(2.4) 
Proof.
Every matrix in satisfies . We also have
On the other hand,
Comparing the two matrices completes the lemma. ∎
2.4. Siegel Model
The Siegel model provides a geometric view of the unitary representation and a simpler formula for the Koranyi inversion. We will in fact define two closely related models, the planar Siegel model that views a point as a vector , and the projective Siegel model that views as a point in complex projective space with homogeneous coordinates . We will denote both models by .
We first identify a point with geometric coordinates with the vector
(2.5) 
Note that this is exactly the image of the vector under the unitary transformation . We will say that has planar Siegel coordinates
(2.6) 
The planar Siegel model of is the set of points in of the form (2.6).
Sometimes, a unitary transformation will take to a point that is not of the same form, but can be rescaled to be such. It will therefore be useful to think of vectors up to rescaling, that is, as elements of complex projective space .
Recall that the complex projective plane is the projectivization of , i.e. the set of nonzero vectors up to rescaling by a nonzero complex number. A point in has homogeneous coordinates , welldefined up to rescaling.
We can now define the projective Siegel model of as the set of points in with homogeneous coordinates .
Abusing notation, we will denote both Siegel models by , with the identification . We have the following simple characterization of points in .
Lemma 2.8.
Let be a null point, that is . Then either or .
We denote the closure of in by .
Remark 2.9.
The region bounded by is the Siegel domain. Complex hyperbolic space is defined on this region and has strong connections to the Heisenberg group, see e.g. [2, 5, 9, 10]. In particular, we intend to discuss the relation of Heisenberg continued fractions to geodesic coding in complex hyperbolic space in an upcoming paper, following [16].
Note that the gauge norm is easy to write in the Siegel model:
Lemma 2.10.
Let . Then the gauge norm of is .
Proof.
An element of has the form for some . The gauge norm of is given by . ∎
The gauge distance is defined as . With this in mind, we show:
Lemma 2.11.
In the planar Siegel model, we have
Proof.
We have associated to the matrix
Applying this matrix to the point , we get the vector
Taking the last two coordinates yields the desired formula. ∎
We now study the action of matrices on the Heisenberg group in the Siegel models. General linear matrices act on by acting on the homogeneous coordinates. Since we have by taking , we also obtain an action on .
Lemma 2.12.
Let and . Then acts on as:
Proof.
The point corresponds to a point in with homogeneous coordinates . We then have
To view as a point in , we renormalize so that the first coordinate is 1, and take the remaining two coordinates. ∎
Elements of do not necessarily preserve the set , but the unitary matrices preserve and therefore . In particular, elements of act transitively on while fixing the point . Denote the matrix by .
Lemma 2.13.
acts on by the Koranyi inversion .
Proof.
We compute, for a point in with geometric coordinates and projective Siegel coordinates :
We thus have that under , the geometric coordinates are mapped to , as desired. ∎
2.5. Lattices and Fundamental Domains
Recall that is the set of Heisenberg points with integer coordinates. In the geometric model , we have . In the Siegel model, is the set of points such that . In the unitary model, we have , where the latter denotes the subset of with Gaussian integer coefficients, and is known as the Picard modular group.
Likewise, we will denote by the set of points in with rational coordinates. Recall that the Heisenberg group admits a family of dilation maps given by in the geometric model. The dilation maps are group isomorphisms and satisfy for all and . It is clear that if and only if there is an integer such that .
We are now interested in the structure and geometry of . We record its generators in the geometric model:
Lemma 2.14.
The group is generated by the elements , , and .
As Falbel–Francics–Lax–Parker showed, and are closely linked (see also [19]):
Theorem 2.15 ([4]).
The group is generated by the matrices , , , and the matrix
corresponding to the mapping .
We now discuss fundamental domains for . Recall that a fundamental domain for is a connected set with piecewise smooth boundary whose translates tile essentially without overlap. That is, and implies .
We require a slightly different definition. We require to consist of an open set and some measurable subset of its boundary (which is not necessarily piecewise smooth) such that and implies . We then have:
Lemma 2.16.
Let be a fundamental domain for . Then the map mapping all points of to is welldefined.
The following lemma follows immediately from the definitions:
Lemma 2.17.
The following regions are fundamental domains for :

The unit cube .

The Dirichlet domain , with a choice of excluded boundary points.
Denote the unit sphere in by . For a subset , let denote the supremum of the norms of the points of , and let denote its Lebesgue measure (in the geometric model).
Lemma 2.18.
Every fundamental domain for satisfies . Furthermore, the domains and satisfy .
Proof.
The radius of is easy to compute because behaves similarly to the Euclidean norm. As in the Euclidean case, the norm is maximized by each corner of the cube. We have .
The radius of seems difficult to compute directly, as the boundary of is more complicated (see Figure 3). We will therefore argue indirectly by means of . Let , and choose so that . We then have . This implies that . Now, by definition of , , so we must also have , so . To prove equality, one shows that the point is on the boundary of .
For the volume computation, it is clear that . To compute for an arbitrary fundamental domain , note that Lebesgue measure is preserved by left translation in the Heisenberg group (which acts by shears). Since can be constructed by rearranging measurable pieces of , the two fundamental domains must have the same volume. ∎
Remark 2.19.
Note that we defined with a particular Hermitian form in mind. Different Hermitian forms provide isomorphic Lie groups , but the lattice depends on the choice of the Hermitian form. If two forms are related by an integer change of coordinates, then the associated lattices are equivalent. If the change of coordinates is not integral, the lattices are not isomorphic as groups (even up to finite index), see [12, 14] Nonetheless, in literature one mostly sees mention of the Picard modular group, defined by a Hermitian form equivalent to our .
3. Heisenberg Continued Fractions
Fix a fundamental domain for the group such that (e.g. or in Lemma 2.17). We begin by establishing some notation.
Definition 3.1.
Given an arbitrary sequence of nonzero digits in , we write the associated continued fraction as
(3.1) 
For an infinite sequence , we define , if this limit exists. The goal of this section is to show that the limit does exist in several important cases, and that the computation of may be simplified by using a recursive algorithm.
Definition 3.2.
We associate with :

A “nearestinteger” map , characterized by
Note that selects the nearest integer in the gauge metric exactly if is the Dirichlet domain .

The Gauss map given by
Remark 3.3.
Working with the geometric model, one sees that each axis is preserved by the Gauss map . In particular, the restriction of to each axis is essentially isomorphic to the nearestinteger Gauss map on . The theory of continued fractions we develop likewise restricts to the classical nearestinteger continued fraction theory on the axes.
Definition 3.4.
Given a point , have:

The forward iterates , for each ,

The continued fraction digits , for each ,

The rational approximants , for each .
Because is defined on , the process of defining forward iterates, continued fraction digits, and rational approximants terminates if for some we have . We will characterize the points for which this happens in Theorem 3.10.
More generally, for a point we can take , and obtain the remaining digits of from as before. However, our focus will be on points in .
It is easy to see that, on finite sequences, is the inverse operation to :
Lemma 3.5.
For with a finite sequence, we have .
Remark 3.6.
The operation is defined without reference to a specific fundamental domain . Thus, while we will show that , we do not in general have . Indeed, problems arise when the get too close to the unit sphere.
For example, let