On a generalization of Littlewood's conjecture

We present a class of lattices in R^d (d>= 2) which we call GL-lattices and conjecture that any lattice is such. This conjecture is referred to as GLC. Littlewood's conjecture amounts to saying that Z^2 is GL. We then prove existence of GL lattices by first establishing a dimension bound for the set of possible exceptions. Existence of vectors (GL-vectors) in R^d with special Diophantine properties is proved by similar methods. For dimension d>= 3 we give explicit constructions of GL lattices (and in fact a much stronger property). We also show that GLC is implied by a conjecture of G. A. Margulis concerning bounded orbits of the diagonal group. The unifying theme of the methods is to exploit rigidity results in dynamics and derive results in Diophantine approximations or the geometry of numbers.

Our interpretation of this conjecture, which naturally leads to its generalization to be described bellow, is as follows: Let us denote by N : R 2 → R the function (x, y) t → x · y (where t stands for transpose). Given a vector v = (α, β) t ∈ R 2 , and an integer n, the set of values N takes on the set Z 2 + nv ⊂ R 2 , is {(nα + k)(nβ + ) : k, ∈ Z}. The distance of this set to zero is exactly nα nβ . Let us denote this distance by N (Z 2 + nv). It is clear that this quantity attains arbitrarily small values as n varies. Littlewood's conjecture asserts that the rate of decay is faster then n. To this end we see that Littlewood's conjecture could be restated as saying ∀v ∈ R 2 , inf n =0 |n| N (Z 2 + nv) = 0.
Adopting this view point we may ask if this should be a special property of the lattice Z 2 . We conjecture that in fact any lattice in the plane should satisfy a similar property (see conjecture 1.2 bellow). In this paper we shall use methods from homogeneous dynamics to establish the existence of lattices in the plane which satisfy the generalized Littlewood conjecture and connect this conjecture to the dynamics of the diagonal group on the space of lattices in the plane. The main tool is the deep measure classification theorem obtained by Einsiedler Katok and Lindenstrauss in [EKL]. Applications to Diophantine approximations are presented.
1.2. Notation Results and conjectures. We first fix our notation and define the basic objects to be discussed in this paper. Throughout this paper d ≥ 2 is an integer. Let X d denote the space of d-dimensional unimodular lattices in R d (i.e. of covolume 1) and let Y d denote the space of translates of such lattices. Points of Y d will be referred to as grids, hence for x ∈ X d and v ∈ R d , y = x + v ∈ Y d is the grid obtained by translating the lattice x by the vector v. We denote by π the natural projection (1.1) For each x ∈ X d , we identify the fiber π −1 (x) in Y d with the torus R d /x. This enables us to define an operation of multiplication by an integer n on Y d , i.e. if y = x + v ∈ Y d then ny = x + nv. The main objective of this paper is to discuss the following generalization of Littlewood's conjecture, referred to in this paper as GLC: (1) A grid y ∈ Y d is Littlewood if (1.3) holds.
(3) A vector v ∈ R d is grid-Littlewood if for any x ∈ X d , the grid y = x + v is Littlewood.
Thus conjecture 1.2 could be rephrased as saying that any lattice (resp vector) is grid-Littlewood. Of particular interest are grid-Littlewood lattices and vectors. For example, as explained in the previous subsection, when d = 2, x = Z 2 ∈ X 2 and v = (α, β) t ∈ R 2 , the grid x + v satisfies (1.3), if and only if the numbers α, β satisfy Littlewood conjecture stated above (conjecture 1.1). Hence, in our terminology, Littlewood's conjecture states simply that the lattice Z 2 is grid-Littlewood. In this paper we shall prove existence of both grid-Littlewood lattices and vectors in any dimension d ≥ 2 and give applications to Diophantine approximations. As remarked above, the methods of proof are those of homogeneous dynamics. Let us now turn to describe the relevant group actions.
SL d (R) and its subgroups acts naturally, via the linear action on R d , on the spaces X d , Y d . The projection π from (1.1) commutes these actions. Of particular interest to us will be the action of the group A d of d × d diagonal matrices with positive diagonal entries and determinant one. The action of A d on R d preserves N : R d → R and in turn, As a consequence, the set of exceptions to GLC The main result in this paper is the following: Theorem 1.4. The set of exceptions to GLC, E d , is contained in a countable union of sets of upper box dimension ≤ dim A = d − 1.
We remark that from the dimension point of view, this is the best possible result without actually proving GLC, because of the A d invariance.
In the fundamental paper [EKL], Einsiedler Katok and Lindenstrauss proved that the set of exceptions to Littlewood's conjecture is a countable union of sets of upper box dimension zero. The main tool in their proof is a deep measure classification theorem. The proof of theorem 1.4 is based on the same ideas and techniques and further more goes along the lines of [EK]. The new ingredient in the proof is lemma 3.2. As corollaries of this we get: Corollary 1.5. The set of x ∈ X d (resp v ∈ R d ) that are not grid-Littlewood, is contained in a countable union of sets of upper box dimension ≤ dim A d = d − 1. In particular, almost any lattice (resp vector) is grid-Littlewood.
Corollary 1.6 (cf [EKL] Theorem 1.5). For a fixed lattice x ∈ X d , the set {y ∈ π −1 (x) : y is not Littlewood} is contained in a countable union of sets of upper box dimension zero.
Corollary 1.7. Any set in X d which has positive dimension transverse to the A d orbits, must contain a grid-Littlewood lattice. In particular if the dimension of the closure of an Theorem 1.4 as well as its corollaries are proved in § 3. We remark here that the only proof we know for the existence of grid-Littlewood lattices in dimension 2 and for grid-Littlewood vectors of any dimension, goes through the proof of theorem 1.4. For lattices of dimension d ≥ 3 the situation is different. As will be seen in § 5, for d ≥ 3, one can exploit rigidity results on commuting hyperbolic toral automorphisms proved by Berend in [B], and give explicit constructions of grid-Littlewood lattice (and in fact of lattices which are grid-Littlewood of finite type, see definition 5.1).
1.3. Applications to Diophantine approximations. Recall that for η ∈ R we denote η = min k∈Z |η + k| . We denote by η * the integer defined by the equation We shall prove the following theorems in § 4 Theorem 1.8. In any set J ⊂ [0, 1] of positive dimension, there is a number α with the following property: ∀β, γ ∈ R, there exists a sequence n i ∈ Z such that |n i | → ∞ and lim max { n i γ ; (n i γ) * α + n i β } = 0, lim |n i | n i γ (n i γ) * α + n i β = 0. (1.5) In particular, when J is the set of badly approximable numbers in the unit interval and we choose β = α and let γ be arbitrary or choose γ = α and let β be arbitrary or even choose β = γ = α we get the following immediate corollary Corollary 1.9.
(1) There are badly approximable numbers α ∈ [0, 1] such that ∀γ ∈ R there exists a sequence n i ∈ Z such that |n i | → ∞ and (1.6) (2) There are badly approximable numbers α ∈ [0, 1] such that ∀β ∈ R there exists a sequence n i ∈ Z such that |n i | → ∞ and (3) There are badly approximable numbers α ∈ [0, 1] such that there exists a sequence n i ∈ Z such that |n i | → ∞ and Theorem 1.10. In any set of dimension more than 1 in the plane, there exists a vector (β, γ) t such that for any α ∈ R there exists a sequence n i ∈ Z with |n i | → ∞ such that (1.5) holds. In particular one could choose α = β or α = γ.

1.4.
Interaction with other conjectures. We end this section by noting that GLC is implied by a conjecture of G.A.Margulis (see [Ma2]) which goes back to [CaSD].
We prove at the end of § §2.2 Proposition 1.12. Conjecture 1.11 implies GLC.
Remark: As will be seen, given a lattice x ∈ X d , the richer the dynamics of its orbit under A d , the easier it will be to prove that it is grid-Littlewood. This suggests an explanation to the difficulty of proving that Z 2 is grid-Littlewood (for it has a divergent orbit) as opposed to proving for example that a lattice with a compact or a dense orbit is grid-Littlewood (see Problem 5.13).
Acknowledgments: I would like express my deepest gratitude to my advisers, Hillel Furstenberg and Barak Weiss for their constant help and encouragement. Special thanks are due to Manfred Einsiedler for his significant contribution to the results appearing in this paper. I would also like to express my gratitude to the mathematics department at the Ohio State University and to Manfred Einsiedler, for their warm hospitality during a visit in which much of the research was conducted.

Preparations
We identify X d with the homogeneous space G d /Γ d in the following manner: For g ∈ G d , the coset gΓ d represents the lattice spanned by the columns of g. We denote this lattice byḡ. Y d is identified with G d R d / Γ d Z d similarly, i.e. for g ∈ G d and v ∈ R d , the coset (g, v)Γ d Z d is identified with the gridḡ + v. We endow X d , Y d with the quotient topologies thus viewing them as homogeneous spaces. We define a natural embedding τ : Y d → X d+1 in the following manner: Note that this embedding is proper. G d and its subgroups act on X d , Y d by multiplication from the left. The reader should check that under our identifications, these actions of G d agrees with the linear action on R d , when applied to points of X d , Y d , thought of as subsets of R d . We embed G d in G d+1 (in the upper left corner) thus allowing G d and its subgroups to act on X d+1 as well. Note that the action commutes with τ.

2.2.
Linking dynamics to GLC. The following observation is useful in connection with GLC: ∀y ∈ Y d inf n =0 |nN (ny)| = inf {|N (w)| : w ∈ τ y , w d+1 = 0} . (2.2) Note the multiple use of the symbol N in the above equation. On the left hand side N refers to a function on Y d while on the right hand side, to a function on R d+1 .
(1) If y, y 0 ∈ Y d are such that τ y 0 ∈ A d+1 τ y , then if y 0 is Littlewood then so is y. ( Proof. The proof of (1) follows from (2.2). The proof (2) follows from (1) and the compactness of the fibers of π.
As done in [EKL] we link GLC to the dynamics of the following cone in A d+1 : e has a noncompact closure), then y is Littlewood.
Proof. Recall Mahler's compactness criterion that says that a set C ⊂ X d+1 is bounded, if and only if there is a uniform positive lower bound on the lengths of non zero vectors belonging to points of C. Let us fix the supremum norm on R d and R d+1 . Let y ∈ Y d . Assume that in the orbit A + d+1 τ y there are lattices with arbitrarily short vectors. Given 0 < < 1, there exists a ∈ A + d+1 and w ∈ τ y such that the vector aw is of length less than . In particular N (aw) = N (w) < . We will be through by (2.2) once we justify that w d+1 = 0. Assume w d+1 = 0. It follows that the length of aw is greater than that of w, as A + d+1 expands the first d coordinates. On the other hand, the vector w = (w 1 . . . w d ) t ∈ R d (which has the same length as w) belongs to the lattice π(y). Let denote the length of the shortest nonzero vector in π(y). We obtain a contradiction once < .
Proof of proposition 1.12. Given a grid y ∈ Y d , if A + d+1 τ y is unbounded then by lemma 2.2 we know that y is Littlewood. Assume that A + d+1 τ y ⊂ K for some compact K ⊂ X d+1 . Choose any one parameter semigroup {a t } t≥0 in the cone A + d+1 and let z be a limit point of the trajectory {a t τ y : t ≥ 0} in K. We claim that z has a bounded A d+1 orbit. To see this note that for any a ∈ A d+1 we have that for large enough t's, aa t is in the cone A + d+1 , thus az ∈ K. Assuming conjecture 1.11, we obtain that z has a compact A d+1 orbit. As τ y contains vectors of the form ( * , . . . , * , 0) t which can be made as short as we wish under the action of A d+1 , we see that the orbit A d+1 τ y is unbounded in X d+1 by Mahler's compactness criterion. It follows that z / ∈ A d+1 τ y and hence z ∈ A d+1 τ y \A d+1 τ y . Theorem 1.3 from [LW] states that any orbit closure of A d+1 in X d+1 which strictly contains a compact A d+1 orbit, is homogeneous. More precisely, there exists a closed group H < G d+1 , strictly containing A d+1 such that A d+1 τ y = Hz. Such a group H must contain a group of the form {u ij (t)} t∈R for some 1 ≤ i = j ≤ d + 1, where u ij (t), is the unipotent matrix all of whose entries are zero but the diagonal entries which are equal to 1 and the ij'th entry that is equal to t. It is easy to see that for any > 0 there exist some t such that u ij (t)z contains a vector v with N (v) ∈ ( , 2 ). Since u ij (t)z ∈ A d+1 τ y , we deduce that τ y contains a vector w with N (w) ∈ ( , 2 ). We deduce that w d+1 = 0 and as was arbitrary, (2.2) implies that y is Littlewood as desired.
2.3. Dimension and entropy. Let us recall the notions of upper box dimension and topological entropy. Let (X, d) be a compact metric space. For any > 0 we denote by S = S (X) the maximum cardinality of a set of points in X with the property that the distance between any pair of distinct points in it is greater or equal to (such a set is called − separated). We define the upper box dimension of X to be Since this is the only notion of dimension we will discuss, we shall denote it by dim X. If we denote by N = N (X) the minimum cardinality of a cover of X by sets of diameter less than , then we also have that dim If we have a continuous map a : X → X, then for > 0, n ∈ N we denote by S n, = S n, (X, a) the maximum cardinality of a set S ⊂ X with the property that for any pair of distinct points x, y ∈ S there exist some 0 n .

Metric conventions and a technical lemma.
For a metric space (X, d) we denote by B X (p), the closed ball of radius around p. If X is a group B X = B X (p) where p is the trivial element (zero or one according to the structure). Given Lie groups G, H... we denote their Lie algebras by the corresponding lower case Gothic letters g, h... Let G be a Lie group. We choose a right invariant metric d(·, ·) on it, coming from a right invariant Riemannian metric. Let Γ < G be a lattice in G. We denote the projection from G to the quotient X = G/Γ, by g →ḡ. We define the following metric on X (also denoted by Under these metrics, for any compact set K ⊂ X there exist an isometry radius (K), i.e. a positive number such that for any x ∈ K, the map g → gx is an isometry between has the identity map as its derivative at zero. It follows that it is bi-Lipschitz on a ball of small enough radius around zero. We refer to such a map as a decomposition chart and to the corresponding radius as a bi-Lipschitz radius. When taking into account the above, we get that given a compact set K ⊂ X and a decomposition g = ⊕ l 1 V i , we can speak of a bi-Lipschitz radius δ(K), for K with respect to this decomposition chart, i.e. we choose δ = δ(K) to be small enough so that the image of B g δ under the decomposition chart will be contained in the ball of radius (K) around the identity element. Note that under these conventions a bi-Lipschitz radius for K with respect to a decomposition chart is always an isometry radius.
Let G be semisimple and R-split (for our purpose it will be enough to consider G = SL d (R)). Let A < G be a maximal R-split torus in G (for example the group of diagonal matrices in SL d (R)). We fix on g a supremum norm with respect to a basis of g whose elements belong to one dimensional common eigenspaces of the adjoint action of A. For an element a ∈ A we denote by U ± (a), u ± (a), the stable and unstable horospherical subgroups and Lie algebras associated with a. That is We denote by u 0 (a) the Lie algebra of the centralizer of a, that is {X ∈ g : Ad a (X) = X}. Note that from the semisimplicity of a, it follows that g = u + (a) ⊕ u 0 (a) ⊕ u − (a). When a fixed element a ∈ A is given, we denote for X ∈ g, its components in u + , u − , u 0 , by X + , X − , X 0 , respectively.
We shall need the following lemma, the reader is advised to skip it for the time being and return to it after seeing it in use in the next section: Proof. Let η > 0 be a bi-Lipschitz radius for the decomposition charts exp and ϕ, corresponding respectively to the trivial decomposition and the decomposition g δ 1 , i = 1, 2 are fixed, we simplify our notation and write instead of Let us formulate two claims that we will use: Claim 1 : There exist 0 < δ 2 < δ 1 and 0 < M, c 1 , such that (2.7) Claim 2 : There exist 0 < δ 3 < δ 2 , such that if v ∈ B g η and k ∈ N are such that Let us describe how to conclude the lemma from these claims: Let λ be the minimum amongst the absolute values of the eigenvalues of Ad a that are greater than 1. Choose δ = δ 3 as in claim 2, M > 0 as in claim 1 and c = c 1 · c 2 , where c 1 is as in claim 1 and Note that because of the choice of the norm on g , for v ∈ g one has for any integer j Let X i , Y i and k ∈ N be as in the statement of the lemma. For any 0 ≤ j ≤ k we have: We used the right invariance of the metric in the first equality, the fact that δ < δ 1 in the second and the relation aϕ(·)a −1 = ϕ(Ad a (·)) in the third. In the last row of inequalities we used claim 2 and the choice of c 2 in the first inequality, (2.8) in the second and claim 1 in the third. We now turn to the proofs of the above claims.
Proof of Claim 1. We use the notation of lemma 2.3. Let 0 < ρ < δ 1 be such that the map (v 1 , v 2 ) → exp v 1 exp −v 2 , takes B g ρ 2 into exp B g η . Since η was chosen to be a bi-Lipschitz radius for the map exp, there is a smooth function w : B g ρ 2 → B g η which satisfies the relation Let us sketch the line of proof we shall pursue: We show that w(v 1 , v 2 ) ∼ r ||v 1 − v 2 ||, for some r > 0. We choose the constants carefully in such a way that given X i , Y i as in the statement of the claim, then there exist a v ∈ g of length less than half of ||X 1 − X 2 ||, with ϕ(v) = exp Ad exp X 2 w(Y 1 , Y 2 ). It then follows from (2.9) that It then follows that (ignoring constants that will appear) Let us turn now to the rigorous argument. The fact that η is a bi-Lipschitz radius for exp implies the existence of a constant r > 0, such that ∀v (2.10) Let M 0 bound from above the operator norm of Ad exp v as v ranges over B g ρ . In (2.9), we have (2.11) Let 0 < δ 2 < ρ be such that 2δ 2 r 2 < ρ. This implies by (2.10), that ∀v 1 , v 2 ∈ B g δ 2 , ||w(v 1 , v 2 )|| < ρ. There exist some 0 < ρ < η such that exp (B g ρ ) ⊂ ϕ(B g ρ ). Note that from the fact that exp is bi-Lipschitz on B g ρ and ϕ −1 is bi-Lipschitz on exp B g ρ , it follows that there exist a constantr such that ∀w ∈ B g ρ , ||w|| ∼r ||ϕ −1 (exp(w)) ||. (2.12) and by our choice of M ||Ad exp X 2 w(Y 1 , Y 2 )|| < ρ .
(2.15) Note that by (2.12) with w = Ad exp X 2 w(Y 1 , Y 2 ), by (2.13), and by the choice of M (2.16) The expressions in (2.6) and in (2.9) are equal to As remarked above, the fact that X i ∈ u + implies that w(X 1 , X 2 ) ∈ u + . From our choice of ρ and the fact that v + , w(X 1 , X 2 ) ∈ B u + ρ , it follows that w (w(X 1 , X 2 ), −v + ) ∈ u + is defined and satisfies: (2.18) Because w(·, ·) takes values in B g η , and ϕ is injective on B g η , it follows from (2.17), (2.18), that u + = w w(X 1 , X 2 ), −v + . Because of (2.10) and (2.16) Thus claim 1 follows with the above choices of δ 2 and M and with c 1 = r 4 2 . Proof of Claim 2. Let M 1 = ||Ad a ||. Let 0 < δ 3 < δ 2 satisfy . (2.19) Let v ∈ B g η and k ∈ N satisfy the assumptions of claim 2. Assume by way of contradiction that there exist some 0 .
This contradicts the assumption that ϕ (Ad j a (v)) ∈ B G δ 3 and (2.19) because ϕ is injective on B g η .

The set of exceptions to GLC
In this section we prove theorem 1.4 and its corollaries. We go along the lines of §4 in [EK] and the fundamental ideas appearing in [EKL]. The main hidden tool is the measure classification theorem in [EKL]. What prevents us from citing known results is the fact that in the embedding τ : Y d → X d+1 (2.1), the grids which are not Littlewood and thus have bounded A + d+1 orbit, do not lie (locally) on single unstable leaves of elements in the cone, but lie on products of unstable leaves (see lemmas 3.1, 3.2).
Beginning of proof of theorem 1.4. Let M i be an increasing sequence of compact sets in X d+1 such that X d+1 = ∪ i M i . Denote F i = y ∈ Y d : A + d+1 τ y ⊂ M i . Lemma 2.2 implies that the set of exceptions to GLC satisfies (3.1) F i is a compact subset of Y d . We shall prove that dim F i ≤ d − 1 and conclude the proof of the theorem. Denote L i = τ (F i ). As τ is, locally, a bi-Lipschitz map, it will be enough to prove dim L i ≤ d − 1. As L i ⊂ τ (Y d ), we wish to describe local neighborhoods in τ (Y d ) in a convenient way. To do this we take Ω ⊂ A d to be a compact symmetric neighborhood of the identity and we denote Given δ > 0 and x ∈ τ (Y d ) ⊂ X d+1 , the set Ω exp B V 1 δ exp B V 2 δ x is a compact neighborhood of x in τ (Y d ). As L i is compact, we can cover it by finitely many sets of the form Thus, the theorem will follow once we prove that as Ω is of dimension d − 1. This is the content of corollary 3.3 bellow. In order to prove this corollary, we need to prove two lemmas. We will return and conclude the proof of theorem 1.4 after proving corollary 3.3.
The following two lemmas furnish the link between dimension and entropy in our discussion. Lemma 3.1 is essentially lemma 4.2 from [EK]. For the reader's convenience and the completeness of our presentation, we include the proof in the appendix. Lemma 3.2 is one of the new ingredients appearing in this paper. For convenience we shall use the following notation: Given a semigroup C ⊂ A d+1 and a compact set K ⊂ X d+1 we denote (3.5) Note that K C is a compact (possibly empty) C-invariant set.
Lemma 3.1. Let C ⊂ A d+1 be a semigroup, a ∈ C and K ⊂ X d+1 a compact set. If for some δ > 0 and x ∈ K dim exp B u + (a) δ · x ∩ K C > 0 then a acts with positive topological entropy on K C .
For the proof of theorem 1.4 we shall need the following generalization of lemma 3.1: Lemma 3.2. Let C 2 ⊂ C 1 ⊂ A d+1 be semigroups, a i ∈ C i , i = 1, 2, and K ⊂ X d+1 a compact set. Assume that there exists subspaces V i of u + (a i ) such that for any b ∈ C 2 , V 1 ⊂ u − (b). Then, there exists δ > 0, such that if for some x ∈ K dim exp B V 1 δ exp B V 2 δ · x ∩ K C 1 > 0, then either a 1 acts with positive topological entropy on K C 1 , or there exists a compact set K ⊃ K, such that a 2 acts with positive topological entropy onK C 2 .
The following corollary goes along the lines of Proposition 4.1 from [EK].
Corollary 3.3. Let C 2 ⊂ C 1 ⊂ A d+1 be open cones, a i ∈ C i , i = 1, 2, and K ⊂ X d+1 a compact set. Assume that there exists subspaces V i of u + (a i ) such that for any b ∈ C 2 , V 1 ⊂ u − (b). Then, there exists δ > 0, such that for any Proof. In the proof of Proposition 4.1 from [EK], it is shown that there cannot be an open cone C ⊂ A d+1 that acts on a compact invariant subset of X d+1 such that some element in C acts with positive topological entropy. Taking δ to be as in lemma 3.2, we see that positivity of the dimension in (3.6) leads to a contradiction.

Remark:
The highly non trivial part of the proof of theorem 1.4 is hidden in the proof of corollary 3.3. This is the use of the measure classification from [EKL].
Concluding the proof of theorem 1.4. In order to conclude the proof of the theorem we use corollary 3.3 with the following choices of K, C i , V i and a i , i = 1, 2 to fit our needs in (3.4): We take V 1 , V 2 to be as in (3.2) and . . , d + 1, 1/(d + 1)!) , a 2 = diag (d + 1, . . . , 2, 1/(d + 1)!) . Note that V 2 = u(a 2 ) + . Moreover, V 1 ⊂ u − (a 2 ), thus we can choose C 2 to be an open cone containing a 2 and contained in C 1 , such that for any b ∈ C 2 , V 1 ⊂ u − (b). Finally we take K = ΩM i .
Proof of lemma 3.2. Note that from the fact that V 1 ⊂ u − (a 2 ) it follows that the sum V 1 + V 2 is direct. Let V 3 be any subspace of g d+1 such that g d+1 = V 1 ⊕ V 2 ⊕ V 3 . Choose δ = δ(K) to be a bi-Lipschitz radius for K with respect to the above decomposition (see § § 2.4 for notation). Assume also that δ satisfies the conclusion of lemma 2.3 with a = a 1 . For the sake of brevity we denote Since δ is a bi-Lipschitz radius for K (with respect to the decomposition V 1 ⊕V 2 ⊕V 3 ), (3.7) implies that the dimension of is positive. From the choice of the norm on g d+1 (see § § 2.4) and from the assumption that for any b ∈ C 2 , V 1 ⊂ u − (b), it follows that for any X ∈ V 1 , ||Ad b (X)|| ≤ ||X||. Choose a compact setK ⊃ exp(B g d+1 δ )K. Denote by π 2 the projection from B 1 × B 2 to B 2 . There are two cases: Case 1: Assume dim π 2 (F ) > 0. We claim that exp (π 2 (F )) x ⊂K C 2 . To see this, note that if Y ∈ π 2 (F ) then there exists some X ∈ B 1 such that exp X exp Y x ∈ K C 1 , and so for any b ∈ C 2 we have Now exp (π 2 (F )) x ⊂ exp B 2 · x ∩K C 2 and therefore, positivity of the dimension of π 2 (F ), implies the positivity of the dimension of exp B 2 · x ∩K C 2 . We apply lemma 3.1 and conclude that a 2 acts with positive topological entropy onK C 2 . Case 2: Assume dim π 2 (F ) = 0 and let us denote dim F = 3ρ with ρ > 0. We will show that a 1 acts with positive topological entropy on K C 1 . Recall the notation of lemma 2.3 (applied to a 1 ). We shall find for arbitrarily large integers n, finite sets S n ⊂ F with the following properties: • For any pair of distinct points (X i , Y i ) ∈ S n , i = 1, 2 (3.8) Given two distinct points in S n , (X i , Y i ), i = 1, 2, let us analyze the rate at which exp X i exp Y i x drift apart from each other under the action of powers of a 1 . For any j ≥ 0 we have that a j 1 exp X i exp Y i x ∈ K by the definition of F , and so, if the distance between these two points is less than δ (which is also an isometry radius for K), we have (3.9) By lemma 2.3, for any k such that for all 0 ≤ j ≤ k, the expressions in (3.9) are smaller than δ, we have In particular, if we set 0 = min{c, δ} then we must have some 0 ≤ j ≤ n for which d(a j 1 exp X 1 exp Y 1 x, a j 1 exp X 2 exp Y 2 x) > 0 . This means that {exp X exp Y x : (X, Y ) ∈ S n } is an (n, 0 )-separated set for (K C 1 , a 1 ). From here, it is easy to derive the positivity of the entropy by the bound we have on the size of S n : To build the sets S n with the above properties, for arbitrarily large n's, we argue as follows: By definition of the dimension one can find a sequence k 0 such that Choose n k ∞ such that λ −n k ≤ k < λ −n k +1 . It follows that (3.10) On the other hand, because we assume dim π 2 (F ) = 0, for any large enough n log(N λ −n M (π 2 (F ))) log(λ n M ) < ρ.
(π 2 (F )) and let E (k) i , i = 1 . . . N k be a covering of π 2 (F ) by subsets of B 2 of diameter less than λ −n k M . Since N k < λ n k ρ M ρ , by (3.10) and the pigeon hole principle, there must exist some 1 ≤ i k ≤ N k with (3.12) Define S n k to be a maximal λ −n k -separated set in π −1 2 (E (k) i k ) ∩ F . By construction, S n k has the desired properties.
Proof of corollary 1.5. The projection π : Y d → X d cannot increase dimension and therefor π(E d ), the set of lattices which are not grid-Littlewood is a countable union of sets of upper box dimension ≤ d − 1. Denote by p : It is bi-Lipschitz with a countable fiber and so if we denote by p 2 : X d × R d → R d the natural projection, then p 2 (p −1 (E d )), the set of vectors which are not grid-Littlewood, is a countable union of sets of upper box dimension ≤ d − 1.
Proof of corollary 1.6. Assume by way of contradiction that there exist x ∈ X d with It follows that if Ω ⊂ A d is a compact neighborhood of the identity, then has dimension greater than dim A d = d − 1. A contradiction to theorem 1.4.
Proof of corollary 1.7. Positivity of the dimension of a subset L ⊂ X d , transverse to the A d orbits, means that A d L contains a compact set of dimension greater then dim A d = d − 1. By theorem 1.4, such a set must contain a grid-Littlewood lattice. If the dimension of the A d orbit closure of a lattice x ∈ X d is greater then d−1, then it contains a grid-Littlewood lattice. It now follows from proposition 2.1, that x is grid-Littlewood.
Remark: The proof of theorem 1.4 gives a bit more than what is stated in its statement. It shows that the set y ∈ Y d : A + d+1 τ y is bounded is a countable union of compact sets of upper box dimension ≤ d − 1. The corollaries of theorem 1.4 have corresponding versions as well.
4. Proof of theorems 1.8, 1.10 Proof of theorem 1.8. It follows from corollary 1.7 and the remark at the end of §3 that there exists α ∈ J such that the lattice x, spanned by the vectors (1, 0) t , (α, 1) t in the plane, is grid-Littlewood and moreover that for any grid y ∈ π −1 (x) one has that A + 3 τ y is unbounded in X 3 . We now untie the definitions and translate this information to Diophantine information on α. Given any vector v = (β, γ) t ∈ R 2 , if we denote y = x+v, the fact that A + 3 τ y is unbounded is equivalent (in light of Mahlerl's compactness criterion), to the existence of a sequence a i ∈ A + 3 going to infinity (i.e. leaving any compact subset) and a sequence of vectors 0 = w i ∈ τ y , such that a i w i → 0 in R 3 . Denote where k i , i , n i ∈ Z, not all equal to zero for each i. Then max e t i |k i + i α + n i β| ; e s i | i + n i γ| ; e −(t i +s i ) |n i | → 0. (4.1) As, t i , s i ≥ 0, it follows from (4.1), first that i = (n i γ) * and than that max { n i γ ; (n i γ) * α + n i β } → 0.
(4.2) Now, taking the product of the three quantities in (4.1) we get We are left to justify why we can choose the n i 's so that |n i | → ∞. Note that if γ is rational and 1, α, β are linearly dependent over Q, then the statement of the theorem is clear. Assume that |n i | is bounded. We might as well assume its constant. Then i is constant too and moreover n i γ = − i , so γ is rational. It then follows from (4.2) that 1, α, β are linearly dependent over Q which concludes the proof.
Proof of theorem 1.10. From corollary 1.5 we get that any set in the plane which is of dimension > 1 must contain a grid-Littlewood vector v = (β, γ) t . Moreover by the remark at the end of §3, we may assume that for any lattice x ∈ X 2 , if we denote y = x + v ∈ Y 2 , then A + 3 τ y is unbounded in X 3 . Given α ∈ R, we apply this to the lattice spanned by (1, 0) t , (α, 1) t and continue as in the proof of theorem 1.8 above.

Lattices that satisfy GLC
In this section we shall explicitly build grid-Littlewood lattices in R d for d ≥ 3. In fact these lattices will possess a much stronger property, namely: Definition 5.1. A grid y ∈ Y d is Littlewood of finite type if there exists a non zero integer n such that N (ny) = 0. A lattice x ∈ X d is grid-Littlewood of finite type if any y ∈ π −1 (x) is Littlewood of finite type.
Definition 5.2. A grid y ∈ Y d is rational if y is an element of finite order in the group π −1 (π(y)) = R d /π(y).
The following list of observations is left to be verified by the reader.
(1) The set of Littlewood of finite type grids is A d invariant.
(2) If y, y 0 ∈ Y d , y 0 ∈ A d y and y 0 is Littlewood of finite type then y is Littlewood of finite type too. (3) If x, x 0 ∈ X d , x 0 ∈ A d x and x 0 is grid-Littlewood of finite type then x is grid-Littlewood of finite type too. (4) Any rational grid is Littlewood of finite type. (5) If x 1 ∈ X d 1 is grid-Littlewood of finite type and x 2 ∈ X d 2 is any lattice, then x 1 ⊕ x 2 ∈ X d 1 +d 2 is grid-Littlewood of finite type. (6) If x 1 , x 2 ∈ X d are such that x 1 is grid-Littlewood of finite type and there exist some c > 0 such that cx 1 is commensurable with x 2 then x 2 is grid-Littlewood of finite type. (7) The standard lattice Z d is not grid-Littlewood of finite type. In fact, for any vector v ∈ R d , all of whose coordinates are irrationals we have that Z d +v is not Littlewood of finite type.
For x ∈ X d , denote by A d,x its stabilizer in A d . Note that A d,x acts on the torus π −1 (x) as a group of automorphisms. From (2) and (4) of proposition 5.3, we deduce the following Lemma 5.4. If for any grid y ∈ π −1 (x), A d,x y ⊂ π −1 (x) contains an Littlewood of finite type grid then x is grid-Littlewood of finite type. In particular, if for any grid y ∈ π −1 (x), A d,x y contains a rational grid then x is grid-Littlewood of finite type.
Recall that a group of automorphisms of a torus π −1 (x) (x ∈ X d ) is called ID, if any infinite invariant set is dense. The following is a weak version of theorem 2.1 from [B]: Theorem 5.5 (Theorem 2.1 [B]). If the stabilizer A d,x of a lattice x ∈ X d under the action of A d satisfies (1) There exist some a ∈ A d,x such that for any n the characteristic polynomial of a n (which is necessarily over Q) is irreducible.
(3) There exists a 1 , a 2 ∈ A d,x which are multiplicatively independent (that is a k 1 a m 2 = 1 ⇒ k = m = 0). then A d,x is ID.
We now turn to the construction of a family of grid-Littlewood of finite type lattices. Let K be a totally real number field of degree d over Q. The ring of integers of K will be denoted by O K .
(1) A lattice in K is the Z-span of a basis of K over Q.
(2) If Λ is a lattice in K then its associated order is defined as O Λ = {x ∈ K : xΛ ⊂ Λ} .
It can be easily verified that for any lattice Λ in K, O Λ is a ring. Moreover, the units in this ring are exactly O * Λ = {ω ∈ K : ωΛ = Λ}. Dirichlet's unit theorem states the following Theorem 5.7 (Dirichlet's unit theorem). For any lattice Λ in K, the group of units O * Λ is isomorphic to {±1} × Z d−1 .
Let σ 1 . . . σ d be an ordering of the different embeddings of K into the reals. Define ϕ : K → R d to be the map whose i'th coordinate is σ i . If we endow R d with the structure of an algebra (multiplication defined coordinatewise), then ϕ becomes a homomorphism of Q algebras (here we think of the fields Q, R as embedded diagonally in R d ). It is well known that if Λ is a lattice in K, then ϕ(Λ) is a lattice in R d . Let us denote by x Λ the point in X d obtained by normalizing the covolume of ϕ(Λ) to be 1. We refer to such a lattice as a lattice coming from a number field. Because ϕ is a homomorphism We can identify the linear map obtained by left multiplication by a ∈ R d on R d with the usual action of the diagonal matrix whose entries on the diagonal are the coordinates of a. We abuse notation and denote the corresponding matrix by the same symbol. After recalling that the product of all the different embeddings of a unit in an order equals ±1 we get that in fact ϕ(O * Λ ) is a subgroup of the stabilizer of x Λ in the group of diagonal matrices of determinant ±1 (in fact there is equality here but we will not use it). To get back into SL d we replace O * Λ by the subgroup O * Λ,+ of totally positive units (that is units, all of whose embeddings are positive). It is a subgroup of finite index in O * Λ . We conclude that ϕ will map O * Λ,+ into A d,x Λ (using our identification of vectors and diagonal matrices).
Lemma 5.8. If x Λ ∈ X d is a lattice coming from a totally real number field K of degree d ≥ 3, then A d,x Λ is an ID group of automorphisms of π −1 (x Λ ).
Proof. It is enough to check that conditions (1),(2),(3) from theorem 5.5 are satisfied. Condition (2) is trivial. Condition (3) is a consequence of Dirichlet's units theorem and the assumption d ≥ 3. To verify condition (1) we argue as follows: We will show that there exist α ∈ O * K such that for any n, α n generates K (this is enough because O * Λ,+ is of finite index in O * K ). Let F 1 . . . F k be a list of all the subfields of K. If we denote for a subset B ⊂ K √ B = {x ∈ K : ∃n such that x n ∈ B} then we need to show that Thus, Dirichlet's units theorem will imply (5.1) once we prove that O * F is of finite index in O * F . We shall give a bound on the order of elements in the quotient O * F /O * F thus showing that the groups are of the same rank. It is enough to show that there exist some integer n 0 such that if x ∈ K satisfies x n ∈ F for some n then x n 0 ∈ F . Let x ∈ K be such an element. Denote by σ 1 . . . σ r the different embeddings of F into the reals and for any 1 ≤ i ≤ r, denote by σ ij , j = 1 . . . s the different extensions of σ i to an embedding of K into the reals. Thus d = rs and σ ij are all the different embeddings of K into the reals. Note that x n ∈ F if and only if for any 1 ≤ i ≤ r σ i1 (x n ) = · · · = σ is (x n ) i.e. if and only if ( σ ik (x) ) n = 1 for all i, j, k. But since there is a bound on the order of roots of unity in K we are done.
We are now in position to prove Theorem 5.9. Any lattice coming from a totally real number field of degree d ≥ 3 is grid-Littlewood of finite type.
Proof. Let x Λ ∈ X d be a lattice coming from a totally real number field of degree d ≥ 3. Using lemma 5.8 and lemma 5.4 we see that the theorem will follow if we will show that any finite A d,x Λ invariant set in π −1 (x Λ ) contain only rational grids. Assume that y ∈ π −1 (x Λ ) lies in a finite invariant set. It follows that there exist e = a ∈ ϕ(O * Λ,+ ) with ay = y.
Since K is spanned over Q by Λ we see that v is in the Q span of cϕ(Λ) = x Λ and hence y is a rational grid as desired.
As a corollary of the ergodicity of the A d action on X d and proposition 5.3 (3), we get the following (we refer the reader to [Sh] for a stronger result).
Corollary 5.10. Almost any x ∈ X d is grid-Littlewood of finite type for d ≥ 3.
The following result appears for example in [LW]: Theorem 5.11. The compact orbits for A d in X d are exactly the orbits of lattices coming from totally real number fields of degree d.
This gives us the following corollary, which can be derived from proposition 5.3 (3), combined with theorem 5.9. We state it separately because of its interesting resemblance to theorem 1.3 from [LW].
Corollary 5.12. For x ∈ X d (d ≥ 3), if A d x contains a compact A d orbit, then x is grid-Littlewood of finite type.
The reader is referred to the paper [Sh] where a significant strengthening of this corollary is established. Let us end this paper with two open problems which emerge from our discussion.
Problem 5.13. Give an explicit example of a Littlewood lattice in dimension 2. In particular, prove that any lattice with a compact A 2 orbit, is Littlewood.
Problem 5.14. Does there exists two dimensional grid-Littlewood of finite type lattices?

Appendix
Proof of lemma 3.1. Let the notation be as in lemma 3.1. The statement of lemma 2.3 simplifies when one chooses the Y i 's to be zero in the original statement: Lemma 2.3, simplified version: For a fixed element e = a ∈ A d+1 there exist λ > 1 and η, c > 0 such that for any X i ∈ B u + (a) η , i = 1, 2, if for an integer k, for any 0 ≤ j ≤ k d(a j exp X 1 , a j exp X 2 ) < η then for any 0 ≤ j ≤ k d(a j exp X 1 , a j exp X 2 ) ≥ cλ j ||X 1 − X 2 ||.
We apply this lemma for the element a ∈ A d+1 appearing in the statement of lemma 3.1. Let 0 < δ < max {η, δ} be a bi-Lipschitz radius for K, with respect to the chart exp (see § § 2.4 for notation). Cover the compact set exp B u + (a) δ x ∩ K C by finitely many sets of the form exp B u + (a) δ y i ∩ K C , for a suitable choice of points y i ∈ K C . By assumption there exists an i such that dim exp B u + (a) δ y i ∩ K C > 0. Because δ is a bi-Lipschitz radius, we have that the dimension of F = F (δ ) = X ∈ B u + (a) δ : exp Xy i ∈ K C (6.1) is positive. Denote it by 2ρ. By definition of dimension this means that there exists a sequence k 0, and k − separated sets S k ⊂ F , such that |S k | > −ρ k . Let n k ∞ be a sequence such that λ −n k ≤ k < λ −n k +1 . Let X 1 , X 2 ∈ S k be two distinct points. Because δ is also an isometry radius for K, if is an integer such that ∀0 ≤ j ≤ , d(a j exp X 1 y i , a j exp X 2 y i ) < δ , then the simplified version of lemma 2.3, stated above implies that ∀0 ≤ j ≤ δ > d (a j exp X 1 y i , a j exp X 2 y i ) = d (a j exp X 1 , a j exp X 2 ) ≥ cλ j ||X 1 − X 2 || ≥ cλ j k > cλ j−n k .

(6.2)
This means that if 0 = min {δ , c}, then {exp Xy i : X ∈ S k }, is an ( 0 , n k ) − separating set for the action of a on K C . We conclude that Thus we achieve the desired conclusion.