Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces
In this paper, we provide a complete theory of Diophantine approximation in the limit set of a group acting on a Gromov hyperbolic metric space. This summarizes and completes a long line of results by many authors, from Patterson’s classic ’76 paper to more recent results of Hersonsky and Paulin (’02, ’04, ’07). Concrete examples of situations we consider which have not been considered before include geometrically infinite Kleinian groups, geometrically finite Kleinian groups where the approximating point is not a fixed point of any element of the group, and groups acting on infinite-dimensional hyperbolic space. Moreover, in addition to providing much greater generality than any prior work of which we are aware, our results also give new insight into the nature of the connection between Diophantine approximation and the geometry of the limit set within which it takes place. Two results are also contained here which are purely geometric: a generalization of a theorem of Bishop and Jones (’97) to Gromov hyperbolic metric spaces, and a proof that the uniformly radial limit set of a group acting on a proper geodesic Gromov hyperbolic metric space has zero Patterson–Sullivan measure unless the group is quasiconvex-cocompact. The latter is an application of a Diophantine theorem.
Key words and phrases:Diophantine approximation, Schmidt’s game, hyperbolic geometry, Gromov hyperbolic metric spaces.
- 1 Introduction
- 2 Gromov hyperbolic metric spaces
- 3 Basic facts about Diophantine approximation
- 4 Schmidt’s game and McMullen’s absolute game
- 5 Partition structures
- 6 Proof of Theorem 6.1 (Absolute winning of )
- 7 Proof of Theorem 7.1 (Generalization of the Jarník–Besicovitch Theorem)
- 8 Proof of Theorem 8.1 (Generalization of Khinchin’s Theorem)
- 9 Proof of Theorem 9.3 ( has full dimension in )
- A Any function is an orbital counting function for some parabolic group
- B Real, complex, and quaternionic hyperbolic spaces
- C The potential function game
- D Winning sets and partition structures
Four of the greatest classical theorems of Diophantine approximation are Dirichlet’s theorem on the approximability of every point with respect to the function (supplemented by Liouville’s result regarding the optimality of this function), Jarník’s theorem stating that the set of badly approximable numbers has full dimension in (which we will henceforth call the Full Dimension Theorem), Khinchin’s theorem on metric Diophantine approximation, and the theorem of Jarník and Besicovitch regarding the dimension of certain sets of well approximable numbers. As observed S. J. Patterson , these theorems can be put in the context of Fuchsian groups by noting that the set of rational numbers is simply the orbit of under the Fuchsian group acting by Möbius transformations on the upper half-plane . In this context, they admit natural generalizations by considering a different group, a different orbit, or even a different space. Beginning with Patterson, many authors [6, 27, 50, 69, 79, 80, 82, 90, 91, 92] have considered generalizations of these theorems to the case of a nonelementary geometrically finite group acting on standard hyperbolic space for some , considering a parabolic fixed point if has at least one cusp, and a hyperbolic fixed point otherwise. Moreover, many theorems in the literature concerning the asymptotic behavior of geodesics in geometrically finite manifolds [2, 3, 23, 35, 63, 64, 81, 84] can be recast in terms of Diophantine approximation. To date, the most general setup is that of S. D. Hersonsky and F. Paulin, who generalized Dirichlet and Khinchin’s theorems to the case of pinched Hadamard manifolds [46, 47] and proved an analogue of Khinchin’s theorem for uniform trees ; nevertheless, they still assume that the group is geometrically finite, and they only approximate by parabolic orbits.111Although they do not fall into our framework, let us mention in passing Hersonsky and Paulin’s paper , which estimates the dimension of the set of geodesic rays in a pinched Hadamard manifold which return exponentially close to a given point of infinitely often, and the two papers [4, 53], which generalize Khinchin’s theorem in a somewhat different direction than us.
In this paper we provide a far-reaching generalization of these results to the situation of a strongly discrete group of general type acting by isometries on an arbitrary Gromov hyperbolic metric space (so that is a special case), and we consider the approximation of a point in the radial limit set of by an arbitrary point in the Gromov boundary of . Thus we are generalizing
the type of group being considered - we do not assume that is geometrically finite,
the space being acted on - we do not even assume that is proper or geodesic, and
the point being approximated with - we do not assume that is a parabolic or hyperbolic fixed point of .
We note that any one of these generalizations would be new by itself. For example, although the Fuchsian group is a geometrically finite group acting on , the approximation of points in its limit set by an orbit other than the orbit of has not been considered before; cf. Examples 1.26 and 1.43. As another example, we remark that losing the properness assumption allows us to consider infinite-dimensional hyperbolic space as a concrete example of a space for which our theorems have not previously been proven. There are a large variety of groups acting on infinite-dimensional hyperbolic spaces which are not reducible in any way to finite-dimensional examples, see e.g. Appendix A.
In addition to generalizing known results, the specific form of our theorems sheds new light on the interplay between the geometry of a group acting on a hyperbolic metric space and the Diophantine properties of its limit set. In particular, our formulation of the Jarník–Besicovitch theorem (Theorem 7.1) provides a clear relation between the density of the limit set around a particular point (measured as the rate at which the Hausdorff content of the radial limit set shrinks as smaller and smaller neighborhoods of the point are considered) and the degree to which other points can be approximated by it. Seeing how this reduces to the known result of R. M. Hill and S. L. Velani  is instructive; cf. Theorem 7.33 and its proof. The same is true for our formulation of Khinchin’s theorem (Theorem 8.1), although to a lesser extent since we are only able to provide a complete analogue of Khinchin’s theorem in special cases (which are nevertheless more general than previous results). Although our formulation of the Full Dimension Theorem (Theorem 5.10) is more or less the same as previous results (we have incorporated the countable intersection property of winning sets into the statement of the theorem), the geometrically interesting part here is the proof. It depends on constructing Ahlfors regular subsets of the uniformly radial limit set with dimension arbitrarily close to the Poincaré exponent (Theorem 5.9); in this way, we also generalize the seminal theorem of C. J. Bishop and P. W. Jones  which states that the radial and uniformly radial limit sets and each have dimension . Theorem 5.9 will appear in a stronger form in [26, Theorem 1.2.1], whose authors include the second- and third-named authors of this paper.222Let us make it clear from the start that although we cite the preprint  frequently for additional background, the results which we quote from  are not used in any of our proofs.
Finally, as an application of our generalization of Khinchin’s theorem we show that for any group acting on a proper geodesic hyperbolic metric space, the uniformly radial limit set has zero Patterson–Sullivan measure if and only if the group is not quasiconvex-cocompact (Proposition 8.18). In the case where and is geometrically finite, this follows from the well-known ergodicity of the geodesic flow [85, Theorem 1], see also [75, Théorème 1.7].
Convention 1. In the introduction, propositions which are proven later in the paper will be numbered according to the section they are proven in. Propositions numbered as 1.# are either straightforward, proven in the introduction, or quoted from the literature.
Convention 2. The symbols , , and will denote asymptotics; a subscript of indicates that the asymptotic is additive, and a subscript of indicates that it is multiplicative. For example, means that there exists a constant (the implied constant), depending only on , such that . means that there exist constants so that . In general, dependence of the implied constant(s) on universal objects such as the metric space , the group , and the distinguished points and will be omitted from the notation.
Convention 3. The notation means as . The notation means
and similarly for .
Convention 4. Let be a geometrically finite subgroup of for some . In the literature, authors have often considered “a parabolic fixed point of if such a point exists, and a hyperbolic fixed point otherwise”. To avoid repeating this long phrase we shall call such a point a conventional point.
Convention 5. The symbol will be used to indicate the end of a nested proof.
Acknowledgements. The first-named author was supported in part by the Simons Foundation grant #245708. The third-named author was supported in part by the NSF grant DMS-1001874. The authors thank two referees who gave valuable suggestions and comments on earlier versions of this paper.
1.1.1. Visual metrics
where and , and where denotes the Gromov product (see Definition 2.1). For sufficiently close to , such a metric exists for all (Proposition 2.15). We remark that when is the ball model of , , and , then we have and we may take
(See Proposition 2.14 below.)
1.1.2. Strongly discrete groups
Let be a metric space, and let .333Here and from now on denotes the group of isometries of a metric space . The notation means that is a subgroup of . There are several notions of what it means for to be discrete, which are inequivalent in general but agree in the case ; see [26, §5] for details. For the purposes of this paper, the most useful notion is strong discreteness.
A group is strongly discrete if for every , the set
is finite, where is a distinguished point.444Here and from now on denotes the closed ball centered at of radius .
Strongly discrete groups are known in the literature as metrically proper. However, we prefer the term “strongly discrete”, among other reasons, because it emphasizes that the notion is a generalization of discreteness.
1.1.3. Groups of general type
An isometry of a hyperbolic metric space is loxodromic if it has two fixed points and in , which are attracting and repelling in the sense that for every neighborhood of , the iterates converge to uniformly on , and if has the same property.
As in the case of , we say that a group is nonelementary if its limit set (see Definition 2.37) contains at least three points.666This definition is not equivalent to the also common definition of a nonelementary group as one which does not preserve any point or geodesic in the compactification/bordification ; rather, a group satisfies this second definition if and only if it is of general type. Since the fixed points of a loxodromic isometry are clearly in the limit set, we have:
Every group of general type is nonelementary.
The converse, however, is not true; there exists groups which are nonelementary and yet are not of general type. Such a group has a unique fixed point on the boundary of , but it is not the only point in the limit set. A good example is the subgroup of generated by the isometries
A group which is nonelementary but not of general type is called a focal group.777See footnote 5.
Proposition 1.6 ([26, Proposition 6.4.1]).
Every nonelementary strongly discrete group is of general type, or in other words, no strongly discrete group is focal.
Due to Proposition 1.6, in our standing assumptions below we could replace the hypothesis that is of general type with the hypothesis that is nonelementary. We have elected not to do so since we would prefer this paper to be as self-contained as possible.
If is a group of general type then has no global fixed points; i.e. for every there exists such that .
1.1.4. Standing assumptions
In the statements below, we will have the following standing assumptions:
is a hyperbolic metric space
is a distinguished point
is a parameter close enough to to guarantee the existence of a visual metric
is a strongly discrete group of general type.
If for some , then we will moreover assume that , , and . Similarly, if , we will assume that , , and .
1.2. The Bishop–Jones theorem and its generalization
Before stating our main results about Diophantine approximation, we present the Bishop–Jones theorem, a theorem which is crucial for understanding the geometry of the limit set of a Kleinian group, and its generalization to hyperbolic metric spaces.
Theorem 1.9 (C. J. Bishop and P. W. Jones, ).
Fix , and let be a nonelementary discrete subgroup of . Let be the Poincaré exponent of , and let and be the radial and uniformly radial limit sets of , respectively. Then
The following theorem is a generalization of the Bishop–Jones theorem:
The “moreover” clause is new even in the case which Bishop and Jones considered, demonstrating that the limit set can be approximated by subsets which are particularly well distributed from a geometric point of view. It does not follow from their theorem since it is possible for a set to have large Hausdorff dimension without having any closed Ahlfors regular subsets of positive dimension (much less full dimension); in fact it follows from the work of D. Y. Kleinbock and B. Weiss  that the set of well approximable numbers forms such a set.999It could be objected that this set is not closed and so should not constitute a counterexample; however, since it has full measure, it has closed subsets of arbitrarily large measure (which in particular still have dimension 1).
As an application of Theorem 5.9, we deduce the following result:
Fix , let , and let be a discrete group acting irreducibly on . Then
where denotes the set of badly approximable vectors in .
For the definitions and proof see Section 9.
1.3. Dirichlet’s theorem, generalizations, and optimality
The most fundamental result in Diophantine approximation is the following:
Theorem 1.13 (Dirichlet 1842).
For every irrational ,
Dirichlet’s theorem can be viewed (up to a multiplicative constant) as the special case of the following theorem which occurs when
Theorem 3.1 (Generalization of Dirichlet’s Theorem).
Let be as in §1.1.4, and fix a distinguished point . Then for every , there exists a number such that for every ,
Here denotes the -radial limit set of (see Definition 2.37).
Furthermore, there exists a sequence of isometries in such that
and such that (3.1) holds with for all .
In some sense, it is not surprising that a point in the radial limit set should be able to be approximated well, since by definition a radial limit point already has a sequence of something resembling “good approximations”, although they are in the interior rather than in the boundary. (Indeed, radial limit points are sometimes called points of approximation.) The proof of Theorem 3.1 is little more than simply taking advantage of this fact. Nevertheless, Theorem 3.1 is important because of its place in our general framework; in particular, its optimality (Theorem 5.10 below) is not obvious.
The orbit of under is precisely the extended rationals . For each and for each in reduced form with we have
It is well-known and easy to check that the Ford circles together with the set constitute a disjoint -invariant collection of horoballs. (Here a subscript of e denotes the Euclidean metric on .) Now note that
where denotes the Hausdorff metric induced by the hyperbolic metric on , and is the stabilizer of in . Thus for all
Exponentiating finishes the proof. ∎
In the special case (1.3), the visual metric is equal to the spherical metric , and is therefore bi-Lipschitz equivalent to the Euclidean metric on compact subsets of .
Apply the Cayley transform to (1.1) above. ∎
The radial limit set of is precisely the set of irrational numbers; in fact for all sufficiently large.
The proof of (GF1 GF2) in  shows that for any geometrically finite group , there exists such that , where is the set of (bounded) parabolic points of . (Here denotes a disjoint union.) In particular, when , we have and , so . ∎
Note that in deducing Theorem 1.13 from Theorem 3.1, the numerator of (1.13) must be replaced by a constant.121212This constant does not depend on by Observation 1.17. This sacrifice is necessary since the constant depends on information about which is not available in the general case.
in the case where , is finitely generated, and with no assumption on by S. J. Patterson in 1976 [69, Theorem 3.2].
in the case where , is geometrically finite with exactly one cusp,131313If has more than one cusp, then Stratmann and Velani (and later Hersonsky and Paulin) essentially prove that for every , there exists such that (3.1) holds, where is a maximal set of inequivalent parabolic points. and is a parabolic fixed point by B. O. Stratmann and S. L. Velani in 1995 [82, Theorem 1].
in the case where is a pinched Hadamard manifold, is geometrically finite with exactly one cusp, and is a parabolic fixed point by S. D. Hersonsky and F. Paulin in 2002 [46, Theorem 1.1].141414It should be noted that the metric which Hersonsky and Paulin use, the Hamenstädt metric, corresponds to the visual metric coming from rather than the visual metric coming from a point in ; however, these metrics are locally bi-Lipschitz equivalent on . Moreover, their “depth” function is asymptotic to the expression , by an argument similar to the proof of Observation 1.15 above. Because our setup is only equivalent asymptotically to theirs, it makes no sense for us to be interested in their computation of the exact value of the constant in Dirichlet’s theorem, which of course we do not claim to generalize.
in the case where is a locally finite tree, is geometrically finite, and is a parabolic fixed point by F. Paulin [73, Théorème 1.2]
A natural question is whether the preceding theorems, Dirichlet’s theorem and its generalization, can be improved. In each case we do not consider improvement of the constant factor to be significant; the question is whether it is possible for the functions on the right hand side of (1.13) and (3.1) to be multiplied by a factor tending to zero, and for the theorems to remain true. More formally, we will say that Theorem 3.1 is optimal for if there does not exist a function such that for all , there exists a constant such that
Let be as in §1.1.4, and fix a distinguished point . A point is called badly approximable with respect to if there exists such that
for all . We denote the set of points that are badly approximable with respect to by , and those that are well approximable by . In the special case (1.3), the set is the classical set of badly approximable numbers.
Trivially (using strong discreteness), the existence of badly approximable points in the radial limit set of implies that Theorem 3.1 is optimal.151515In some contexts, a Dirichlet theorem can be optimal even when badly approximable points do not exist; see  for such an example. In fact, we have the following:
Theorem 1.19 (Full Dimension Theorem, Jarník 1928).
, so Theorem 1.13 is optimal.
Theorem 5.10 (Generalization of the Full Dimension Theorem).
We shall prove Theorem 5.10 by proving that the sets are absolute winning in the sense of C. T. McMullen  on the sets described in Theorem 5.9. Our theorem then can be deduced from the facts that the countable intersection of absolute winning sets is absolute winning, and that an absolute winning subset of an Ahlfors -regular set has dimension ((i), (ii), and (iii) of Proposition 4.4 below).
1.3.2. Geometrically finite groups and approximation by parabolic points
In the following, we shall assume that for some , and that is a nonelementary geometrically finite group. We will say that a point is badly approximable by parabolic points if where is a complete set of inequivalent parabolic fixed points. The set of points which are badly approximable by parabolic points admits a number of geometric interpretations:
For a point , the following are equivalent:
is badly approximable by parabolic points.
161616Here and from now on denotes the geodesic segment connecting and .avoids some disjoint -invariant collection of horoballs centered at the parabolic points of .
The image of in the quotient manifold is bounded.
is a uniformly radial limit point of .
The proof of this proposition is not difficult and will be omitted. Certain equivalences have been noted in the literature (e.g. the the equivalence of (B) and (C) was noted in ).
Note that according to Theorem 5.10, the set of points which are badly approximable by parabolic points has full dimension in the radial limit set.
in the case where , is a lattice, and is a finite set of conventional points by Patterson in 1976 [69, the Theorem of §10].
in the case where , is convex-cocompact, and is a hyperbolic fixed point by B. O. Stratmann in 1994 [79, Theorem B].
in the case where , is a lattice, and is a finite set of parabolic points by B. O. Stratmann in 1994 [79, Theorem C].
The case where , is a lattice, and is a finite set of parabolic points can be deduced from [23, Corollary 5.2].
It is likely that the techniques used in  can be used to prove Theorem 5.10 in the case where is a proper geodesic hyperbolic metric space, is any strongly discrete group of general type, and is a set of bounded parabolic points (see Subsection 2.7). However, note that such a proof would depend on our Theorem 5.9; cf. Remark 1.12.
1.4. The Jarník–Besicovitch theorem and its generalization
For any irrational , we can define the exponent of irrationality
Then Dirichlet’s theorem (Theorem 1.13) implies that for every irrational . For each let
be the set of very well approximable numbers and Liouville numbers, respectively.
Theorem 1.23 (Jarník 1929 and Besicovitch 1934).
For every , we have
In particular and .
Let be as in §1.1.4. For any point , we can analogously define
Note that in the special case (1.3), we have , , and .
To formulate our generalization of Theorem 1.23, we need some preparation. Let be as in §1.1.4, and fix a distinguished point . We recall that for and , the Hausdorff content of in dimension is defined as
Let be the Poincaré exponent of , and fix . For each we define
For the remainder of this subsection, we assume that .
if and only if for all .
The function is continuous on .
It is obvious from (7.1) that depends on the function , which in turn depends on the geometry of the limit set of near the point . We will therefore consider special cases for which the function can be computed explicitly.
1.4.1. Special cases
The simplest case for which the function can be computed explicitly is the case where is a radial limit point. Indeed, we have the following:
This example demonstrates that the expression is maximized when is a radial limit point. The intuitive reason for this is that there are a lot of “good approximations” to , so by a sort of duality principle there are many points which are approximated very well by .
We remark that if is of divergence type, then a point in represents a “typical” point; see [26, Theorem 1.4.1]. Thus, generically speaking, the Hausdorff dimension of the set depends only on and , and depends neither on the point nor the group .
Let and let . Example 7.30 shows that for every ,
This does not follow from any known result.
Next we consider the case where is a bounded parabolic point (see Subsection 2.7):
If , then , where is the rank of . Thus (7.43) reduces to
in this case.
Moving beyond the geometrically finite case, let us return to the idea of a duality principle between approximations of a point and the size of the set of points which are well approximable by that point. Another common notion of approximability is the notion of a horospherical limit point, i.e. a point for which every horoball centered at (see Definition 2.33) is penetrated by the orbit of .
In particular, notice that if there is a sequence such that is bounded polynomially in terms of , then . On the other hand,
There exists a (discrete) nonelementary Fuchsian group and a point such that .
History of Theorem 7.1. Although Theorem 7.1 has not been stated in the literature before (we are the first to define the function ), the equations (7.39) and (1.6) have both appeared in the literature. (7.39) was proven191919Recall that (7.39) holds whenever is either a radial limit point (e.g. a hyperbolic fixed point) or a (bounded) parabolic point satisfying .
in the case where , is a lattice, and is a parabolic fixed point by M. V. Melián and D. Pestana in 1993 [64, Theorems 2 and 3].
in the case where , is convex-cocompact, and is a hyperbolic fixed point by M. M. Dodson, M. V. Melián, D. Pestana, and S. L. Velani in 1995 [27, Theorem 3].
in the case where , is geometrically finite and satisfies ,202020Since , the condition is equivalent to the condition that for every parabolic point , i.e. the necessary and sufficient condition for (7.43) to reduce to (7.39) (Remark 1.29). When , Stratmann proves an upper and a lower bound for , neither of which are equal to the correct value (7.43). and is a parabolic fixed point by B. O. Stratmann in 1995 [80, Theorem B].
Additionally, the direction of (7.39) was proven
Finally, (1.6) has been proven in the literature only once before,
1.5. Khinchin’s theorem and its generalization
We now consider metric Diophantine approximation, or the study of the Diophantine properties of a point chosen at random with respect to a given measure.222222In this paper, measures are assumed to be finite and Borel. In this case, we would like to consider a function and consider the points which are well or badly approximable with respect to . The classical definition and theorem are as follows:
An irrational point is -approximable if there exist infinitely many such that
Theorem 1.35 (Khinchin 1924).
Let be any function.
If the series
converges, then almost no point is -approximable.
In both cases the implicit measure is Lebesgue.
Again, we would like to consider a generalization of this theorem to the setting of §1.1.4. We will replace Lebesgue measure in the above theorem by an ergodic -quasiconformal measure (see Definition 2.51), where is the Poincaré exponent of . Sufficient conditions for the existence and uniqueness of such a measure are given in Subsection 2.8. Thus, we will assume that we have such a measure, and that we want to determine the Diophantine behavior of almost every point with respect to this measure.
We will now define our generalization of the notion of a -approximable point:
Let be as in §1.1.4, and fix a distinguished point . Let be a function such that the function is nonincreasing.242424Although this is not the weakest assumption which we could have used here, it has the advantage of avoiding technicalities about what regularity hypotheses on are necessary for which statements; this hypothesis suffices for every statement we will consider. We have added the hypothesis to the definition itself because if a weaker hypothesis is used, then the equation (1.8) may need to be modified. We say that a point is -well approximable if for every there exists such that
We denote the set of all -well approximable points by , and its complement by .
Note that our definition differs from the classical one in that a factor of has been added.252525The reason that we have called points satisfying our definition -well approximable rather than -approximable is to indicate this distinction. This is in keeping with the philosophy of hyperbolic metric spaces, since most quantities are considered to be defined only “up to a constant”. Furthermore, our definition is independent of the chosen fixed point . The relation between our definition and the classical one is as follows:
Let be a function such that the function is nonincreasing. In the case (1.3), a point is -well approximable if and only if for every it is -approximable where .
Theorem 8.1 (Generalization of Khinchin’s Theorem262626The fact that Theorem 8.1 is truly a generalization of Theorem 1.35 is proven in Example 1.42 below.).
Let be as in §1.1.4, and fix a distinguished point . Let be the Poincaré exponent of , and suppose that is an ergodic -quasiconformal probability measure satisfying . Let
Let be a function such that the function is nonincreasing. Then:
The fact that is either or is an easy consequence of the assumption that is ergodic, since is a measurable -invariant set.
(iiB) is not a special case of (iiA) as demonstrated by Proposition 8.15 below. In particular, the proof of (iiB) depends crucially on the fact that if is a parabolic point, then the map is highly non-injective, which reduces the number of potential approximations to consider. In general the map is usually injective, so this reduction cannot be used. Thus, it seems unlikely that the converse of (i) of Theorem 8.1 should hold in general.
1.5.1. Special cases
When considering special cases of Theorem 8.1, we will consider the following three questions:
Under what hypotheses is it possible to simplify (asymptotically) the series (8.1)?
Under what hypotheses can we prove that and/or ?
To answer these questions we consider the following asymptotic formulas:
These asymptotics will not be satisfied for every group and every point , but we will see below that there are many reasonable hypotheses which ensure that they hold. When the asymptotics are satisfied, we have the following: