On a Generalization of Littlewood's Conjecture 1

JOURNAL OF MODERN DYNAMICS doi: 10.3934/jmd.2009.3.457 VOLUME 3, NO. 3, 2009, 457–477

ON A GENERALIZATION OF LITTLEWOOD’S CONJECTURE

URI SHAPIRA (Communicated by Anatole Katok)

d ABSTRACT. We present a class of lattices in R (d 2) which we call grid- ≥ Littlewood lattices and conjecture that any lattice is such. This conjecture is referred to as GLC. Littlewood’s conjecture amounts to saying that Z2 is grid- Littlewood. We then prove the existence of grid-Littlewood lattices by ﬁrst es- tablishing a dimension bound for the set of possible exceptions. The existence of vectors (grid-Littlewood-vectors) in Rd with special Diophantine properties is proved by similar methods. Applications to Diophantine approximations are given. For dimension d 3, we give explicit constructions of grid-Littlewood ≥ lattices (and in fact lattices satisfying 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 ([4, 1, 5]), and derive results in Diophantine approximations or the geometry of numbers.

CONTENTS 1. Introduction 457 2. Preparations 461 3. The set of exceptions to GLC 467 4. Proofs of Theorems 1.8 and 1.10 472 5. Lattices that satisfy GLC 472 6. Appendix 476

1. INTRODUCTION 1.1. Introductory discussion. In this paper, we wish to discuss a generalization of the following well-known conjecture due to Littlewood (see [6]):

CONJECTURE 1.1 (Littlewood). For all α,β R, infn 0 n nα nβ 0, where ∈ 6= | |〈 〉〈 〉 = γ : minn Z γ n for γ R. 〈 〉 = ∈ − ∈ ¯ ¯ ¯ ¯ Received April 17, 2009, revised June 30, 2009. 2000 Mathematics Subject Classiﬁcation: Primary: 37A17, 37N99; Secondary: 11H46, 11J20, 11J25, 11J37. Key words and phrases: Littlewood’s conjecture, diagonal ﬂow, inhomogeneous lattices, product of inhomogeneous linear forms. This work was part of the author’sPh.D thesis at the Hebrew University of Jerusalem.

Our interpretation of this conjecture, which naturally leads to a generalization to be described below, is as follows: let us denote by N : R2 R the function → (x, y)t x y (where t is the transpose). Given a vector v (α,β)t R2, and an 7→ · = ∈ integer n, the set of values N takes on the set Z2 nv R2 is + ⊂ (nα k)(nβ ℓ) : k,ℓ Z . + + ∈ © ª The distance of this set to zero is exactly nα nβ . Let us denote this distance 〈 〉〈 〉 by N Z2 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 can be restated as for all v R2 inf n N(Z2 nv) 0. ∈ n 0| | + = 6= Adopting this viewpoint, we may ask if this should be a special property of the lattice Z2. We conjecture that, in fact, any lattice in the plane should satisfy a similar property (see conjecture 1.2 below). 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, Ka- tok, and Lindenstrauss in [4]. 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 ≥ d integer. Let Xd denote the space of d-dimensional unimodular lattices in R (i.e., of covolume 1), and let Yd denote the space of translates of such lattices. Points of Y will be referred to as grids; hence for x X and v Rd , y x v Y is d ∈ d ∈ = + ∈ d the grid obtained by translating the lattice x by the vector v. We denote by π the natural projection π (1.1) Y X , x v x. d −→ d + 7→ In the next section, we shall see that Xd and Yd are homogeneous spaces of Lie groups and equip them with metrics. In the meantime, the reader should think d of the points of Xd and Yd simply as subsets of R . Let N : Rd R denote the function N(w) d w . For a grid y Y , we write → = 1 i ∈ d Q (1.2) N(y) : inf N(w) : w y . = | | ∈ © ª REMARK. A word of caution: our use of the symbol N is ambiguous in several d respects. It denotes simultaneously functions defined on Yd and R and moreover, the dimension, d, will vary from time to time in our discussion. We choose this notation to be consistent with notation in the literature.

1 d For each x X , we identify the ﬁber π− (x) in Y with the torus R /x. This ∈ d d enables us to deﬁne an operation of multiplication by an integer n on Yd , i.e., if y x v Y , then ny x nv. The main objective of this paper is to discuss = + ∈ d = + JOURNAL OF MODERN DYNAMICS VOLUME 3, NO. 3 (2009), 457–477 ON A GENERALIZATIONOF LITTLEWOOD’S CONJECTURE 459 the following generalization of Littlewood’s conjecture, referred to in this paper as GLC:

CONJECTURE 1.2 (GLC). For any d 2 and y Y , ≥ ∈ d (1.3) inf nN ny 0. n 0 = 6= ¯ ¡ ¢¯ DEFINITION 1.3. ¯ ¯

1. A grid y Yd is Littlewood if (1.3) holds. ∈ 1 2. A lattice x X is grid-Littlewood if any grid y π− (x) is Littlewood. ∈ d ∈ 3. A vector v Rd is grid-Littlewood if for any x X , the grid y x v is ∈ ∈ d = + Littlewood. Thus Conjecture 1.2 could be rephrased as saying that any lattice (respectively 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 Z2 X and v (α,β)t R2, the grid x v satisﬁes (1.3) if and only = = ∈ 2 = ∈ + if the numbers α,β satisfy the Littlewood conjecture stated above (Conjecture 1.1). Hence, in our terminology, Littlewood’s conjecture states simply that the lattice Z2 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. d SLd (R) and its subgroups act naturally, via the linear action on R , on the spaces Xd and Yd . The projection π from (1.1) commutes with these actions. Of particular interest to us will be the action of the group A of d d diagonal d × matrices with positive diagonal entries and determinant one. The action of Ad on Rd preserves N : Rd R and, in turn, N : Y R is A invariant too. As a → d → d consequence, the set of exceptions to GLC (1.4) E y Y : y is not Littlewood d = ∈ d © ª is Ad invariant. The main result in this paper is the following:

THEOREM 1.4. The set of exceptions to GLC, Ed , is containedin a countableunion 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 Ad invariance. In the fundamental paper [4], 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 furthermoregoes along the same lines of [3]. The new ingredient in the proof is Lemma 3.2. As corollaries of Theorem 1.4 we get:

COROLLARY 1.5. Those x X (respectively v Rd ) that are not grid-Littlewood ∈ d ∈ are contained in a countable union of sets of upper box dimension dim A ≤ d = d 1. In particular, almost any lattice (respectively vector) is grid-Littlewood. − JOURNAL OF MODERN DYNAMICS VOLUME 3, NO. 3 (2009), 457–477 460 URI SHAPIRA

1 COROLLARY 1.6 (cf. [4] Theorem 1.5). Foraﬁxedlattice x X ,theset {y π− (x) : ∈ d ∈ y is not Littlewood} is contained in a countable union of sets of upper box dimension zero.

COROLLARY 1.7. Any set in Xd that has positive dimension transverse to the Ad - orbits must contain a grid-Littlewood lattice. In particular,if the dimensionof the closure of an orbit A x (x X ) is bigger than d 1, then x is grid-Littlewood. d ∈ d − Theorem 1.4 and 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 [1] to give explicit constructions of grid- Littlewood lattices (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 ∈ write η : mink Z η k . We denote by η∗ the integer deﬁned by the equation 〈 〉 = ∈ + ¯ ¯ ¯ ¯ η η η∗ . 〈 〉 = + ¯ ¯ 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: for all β,γ R, there exists a sequence n Z such ∈ i ∈ that n and | i | → ∞ (1.5) limmax n γ ; (n γ)∗α n β 0, lim n n γ (n γ)∗α n β 0. 〈 i 〉 〈 i + i 〉 = | i |〈 i 〉〈 i + i 〉 = © ª 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 for all γ R, ∈ ∈ there exists a sequence n Z such that n and i ∈ | i | → ∞ (1.6) limmax n γ ; (n γ)∗ n α 0, lim n n γ (n γ)∗ n α 0. 〈 i 〉 〈 i + i 〉 = | i |〈 i 〉〈 i + i 〉 = © ¡ ¢ ª ¡ ¢ 2. Therearebadlyapproximablenumbers α [0,1] suchthatforall β R, there ∈ ∈ exists a sequence n Z such that n and i ∈ | i | → ∞ (1.7) limmax n α ; (n α)∗α n β 0, lim n n α (n α)∗α n β 0. 〈 i 〉 〈 i + i 〉 = | i |〈 i 〉〈 i + i 〉 = © ª 3. There are badly approximable numbers α [0,1] such that there exists a se- ∈ quence n Z satisfying n and i ∈ | i | → ∞ (1.8) limmax{ n α ; (n α)∗ n α } 0, lim n n α (n α)∗ n α 0. 〈 i 〉 〈 i + i 〉 = | i |〈 i 〉〈 i + i 〉 = ¡ ¢ ¡ ¢ 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 is a sequence n Z with n such ∈ i ∈ | i | → ∞ that (1.5) holds.

JOURNAL OF MODERN DYNAMICS VOLUME 3, NO. 3 (2009), 457–477 ON A GENERALIZATIONOF LITTLEWOOD’S CONJECTURE 461

1.4. Interaction with other conjectures. We note that GLC is implied by a conjecture of G. A. Margulis (see [7]) which goes back to [2].

CONJECTURE 1.11 (Margulis). Any bounded Ad 1-orbit in Xd 1 is compact. + + We will prove the following at the end of §2.2.

PROPOSITION 1.12. Conjecture 1.11 implies GLC.

REMARK. As will be seen, given a lattice x X , the richer the dynamics of its or- ∈ d bit under Ad , the easier it will be to prove that it is grid-Littlewood. This suggests an explanation of the difﬁculty of proving that Z2 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).

2. PREPARATIONS 2.1. X ,Y as homogeneous spaces. Write G : SL (R) and Γ : SL (Z). We d d d = d d = d identify X with the homogeneous space G /Γ in the following manner: for g d d d ∈ Gd , the coset gΓd represents the lattice spanned by the columns of g. We denote this lattice by g¯. Y is identified with G ⋉ Rd Γ ⋉ Zd similarly, i.e., for g d d d ∈ G and v Rd , the coset (g, v)Γ ⋉Zd ¡is identified¢±¡ with the¢ grid g¯ v. We endow d ∈ d + Xd and Yd with the quotient topologies, thus viewing them as homogeneous spaces. We define a natural embedding τ: Yd , Xd 1 in the following manner: → + for any y g¯ v Y = + ∈ d g v (2.1) τy Γd 1 Xd 1. = µ0 1¶ + ∈ +

Note that this embedding is proper. Gd and its subgroups act on Xd ,Yd by multiplication from the left. The reader should check that under our identiﬁcations, d this action of Gd agrees with the linear action on R when applied to points of d Xd and Yd , thought of as subsets of R . We embed Gd in Gd 1 (in the upper left + corner) thus allowing Gd and its subgroups to act on Xd 1 as well. Note that the + action commutes with τ. 2.2. Linking dynamics to GLC. The following observation is useful in connec- tion with GLC: for any y Y ∈ d (2.2) inf nN(ny) inf N(w) : w τy , wd 1 0 . n 0 = | | ∈ + 6= 6= ¯ ¯ © ª Note the multiple use¯ of the symbol¯ N in the above equation. On the left-hand side, N refers to a function on Yd , while on the right-hand side, it refers to a d 1 function on R + .

PROPOSITION 2.1 (Inheritance).

1. If y, y0 Yd are such that τy0 Ad 1τy ,thenif y0 is Littlewood, so is y. ∈ ∈ + 2. If x, x X are such that x A x,thenif x is grid-Littlewood, so is x. 0 ∈ d 0 ∈ d 0 Proof. The proof of (1) follows from (2.2). The proof of (2) follows from (1) and the compactness of the ﬁbers of π.

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As in [4], we link GLC to the dynamics of the following cone in Ad 1: + t1 td 1 (2.3) A+ diag(e ...e + ) Ad 1 : ti 0,i 1...d . d 1 = ∈ + > = + © ª LEMMA 2.2. For y Y ,if A+ τ is unbounded (i.e., has a noncompact closure), ∈ d d 1 y then y is Littlewood. +

Proof. Recall Mahler’s compactness criterion, which says that a set C Xd 1 is ⊂ + bounded if and only if there is a uniform positive lower bound on the lengths of nonzero vectors belonging to points of C. Let us fix the supremum norm on Rd Rd 1 and + . Let y Yd . Assume that in the orbit Ad+ 1τy there are lattices with ar- ∈ + bitrarily short vectors. Given 0 ǫ 1, there exists a A+ and w τ such that < < ∈ d 1 ∈ y the vector aw is of length less than ǫ. In particular, N(aw+ ) N(w) ǫ. In view = < of (2.2), it suffices to argue that wd 1 0. Assume wd 1 0. It follows that the + 6= + = length of aw is greater than that of w, as Ad+ 1 expands the first d coordinates. +t d On the other hand, the vector w ′ (w ... w ) R (which has the same length = 1 d ∈ as w) belongs to the lattice π(y). Let ℓ denote the length of the shortest nonzero vector in π(y). We obtain a contradiction once ǫ ℓ. <

Proof of Proposition 1.12. Given a grid y Yd , if Ad+ 1τy is unbounded, then by ∈ + Lemma 2.2, we know that y is Littlewood. Assume that Ad+ 1τy K for some + ⊂ compact K Xd 1. Choose any one-parameter semigroup {at }t 0 in the cone ⊂ + ≥ Ad+ 1, and let z be a limit point of the trajectory at τy : t 0 in K . We claim that + ≥ z has a bounded Ad 1-orbit. To see this, note© that for anyª a Ad 1, we have + ∈ + for every large enough t that aat is in the cone Ad+ 1; thus az K . Assuming + ∈ Conjecture 1.11, z has a compact Ad 1-orbit. As τy contains vectors of the form t + ( ,..., ,0) which can be made as short as we wish under the action of Ad 1, ∗ ∗ + we see that the orbit Ad 1τy is unbounded in Xd 1 by Mahler’s compactness + + criterion. It follows that z Ad 1τy , and hence z Ad 1τy Ad 1τy . Theorem ∉ + ∈ + à + 1.3 from [5] states that any orbit closure of Ad 1 in Xd 1 which strictly contains a + + compact Ad 1-orbit is homogeneous. More precisely, there exists a closed group + H Gd 1, strictly containing Ad 1, such that Ad 1τy H z. Such a group H must < + + + = contain a group of the form u (t) for some 1 i j d 1, where u (t), is ij t R ≤ 6= ≤ + ij the unipotent matrix all of whose© entriesª ∈ are zero except for the diagonal entries, which are equal to 1, and the ij ’th entry, which is equal to t. It is easy to see that for any ǫ 0, there exists some t such that u (t)z contains a vector v with > ij N(v) (ǫ,2ǫ). Since uij (t)z Ad 1τy , we deduce that τy contains a vector w ∈ ∈ + with N(w) (ǫ,2ǫ). Thus wd 1 0, and as ǫ was arbitrary, (2.2) implies that y is ∈ + 6= 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 ǫ 0, we > denote by S S (X ) the maximum cardinality of a set of points in X with the ǫ = ǫ property that the distance between any pair of distinct points in it is at least ǫ (such a set is called ǫ-separated). We deﬁne the upper box dimension of X to be

logSǫ dimbox X limsup . = ǫ 0 log ǫ → | | JOURNAL OF MODERN DYNAMICS VOLUME 3, NO. 3 (2009), 457–477 ON A GENERALIZATIONOF LITTLEWOOD’S CONJECTURE 463

Since this is the only notion of dimension we will discuss, we shall denote it by dim X . If we denote by Nǫ Nǫ(X ) the minimum cardinality of a cover of X = logNǫ by sets of diameter less than ǫ, then we also have that dim X limsupǫ 0 logǫ . = → Note that if f : X Y is a bi-Lipschitz map, then A X is of zero dimension| | if → ⊂ and only if f (A) Y is. ⊂ If we have a continuous map a : X X , then for ǫ 0,n N we denote by → > ∈ S S (X , a) the maximum cardinality of a set S X with the property that n,ǫ = n,ǫ ⊂ for any pair of distinct points x, y S, there exist some 0 i n with d(ai x, ai y) ∈ ≤ ≤ > ǫ (such a set is called (n,ǫ)-separated for a). The topological entropy of a is de- ﬁned to be

logSn,ǫ htop (a) lim limsup . = ǫ 0 n n →

2.4. Metric conventions and a technical lemma. For a metric space (X ,d), we denote by B X (p) the closed ball of radius ǫ around p. If X is a group, B X B X (p), ǫ ǫ = ǫ 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 g¯. = 7→ We deﬁne the following metric on X (also denoted by d( , )): · · ¯ (2.4) d(g¯,h) inf d(gγ1,hγ2) inf d(g,hγ). = γi Γ = γ Γ ∈ ∈ Under these metrics, for any compact set K X there exists an isometry radius ⊂ ǫ(K ), i.e., a positive number ǫ such that for any x K , the map g g x is an ∈ 7→ isometry between BG and B X (x). Given a decomposition of g l V , the map ǫ ǫ = 1 i v expv ...expv (where v v and v V ) has the identityL map as its de- 7→ 1 l = i i ∈ i rivative at zero. It follows that itP 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, given a compact set K X and a decomposition g l V , we can speak of a ⊂ = 1 i bi-Lipschitz radius, δ(K ), for K with respect to this decompositionL chart, i.e., we g choose δ δ(K ) to be small enough such that the image of B under the decom- = δ position chart will be contained in the ball of radius ǫ(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 (R)). Let A G be a maximal R-split torus in G (for example the group = d < of diagonal matrices in SLd (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) and u±(a) the ∈ stable and unstable horospherical subgroups and Lie algebras associated with a.

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That is,

n n 1 U +(a) g G : a− ga e ,U −(a) U +(a− ), n = n ∈ −−−−→→∞ o = n u±(a) X g : Ad (X ) 0 . = ∈ a −−−−−→n n →∓∞ o 0 We denote by u (a) the Lie algebra of the centralizer of a,thatis{X g : Ada(X ) ∈ 0 = X }. Note that from the semisimplicity of a, it follows that g u+(a) u (a) u−(a). = ⊕ ⊕ When a ﬁxed element a A is given, we denote for X g its components in 0 0 ∈ ∈ u+,u−,u , by X +, X −, X , 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.

LEMMA 2.3. For a fixed element e a A, there exist λ 1 and δ, M,c 0 such g 6= ∈ > X1 X2 > that for any X ,Y B ,i 1,2, with X u+(a) and Y Y k − k , iffor an i i ∈ δ = i ∈ k 1 − 2k < M integer k, for any 0 j k ≤ ≤ d(a j exp(X )exp(Y ), a j exp(X )exp(Y )) δ 1 1 2 2 < then for any 0 j k ≤ ≤ d(a j exp(X )exp(Y ), a j exp(X )exp(Y )) cλj X X . 1 1 2 2 ≥ k 1 − 2k 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 0 g 0 g u+(a) u (a) u−(a); i.e., ϕ: B G is the map ϕ(v) expv+ expv exp v−. = ⊕ ⊕ η → = Let 0 δ η satisfy < 1 < g g (2.5) exp(X1)exp(Y1)exp( Y2)exp( X2) ϕ(Bη) for all Xi ,Yi B ,i 1,2. − − ∈ ∈ δ1 = g 4 g We can define u : B Bη by δ1 → ¡ ¢ 1 g u(Xi ,Yi ) ϕ− exp(X1)exp(Y1)exp( Y2)exp( X2) for all Xi ,Yi B ,i 1,2. = − − ∈ δ1 = ¡ ¢ g When Xi ,Yi B ,i 1,2 are fixed, we simplify our notation and write u,u±, and δ1 0 ∈ = 0 u instead of u(Xi ,Yi ),u±(Xi ,Yi ), and u (Xi ,Yi ). Thus we have the identity: 0 (2.6) ϕ(u) exp(u+)exp(u )exp(u−) exp(X )exp(Y )exp( Y )exp( X ) = = 1 1 − 2 − 2 g for all Xi ,Yi B ,i 1,2. ∈ δ1 = Let us formulate two claims that we will use.

CLAIM 1. There exist 0 δ δ and M,c 0 such that < 2 < 1 1 > (2.7) g X1 X2 if Xi ,Yi B ,i 1,2, Xi u+, Y1 Y2 k − k, then u+ c1 X1 X2 . ∈ δ2 = ∈ k − k < M k k > k − k g CLAIM 2. There exist 0 δ3 δ2 such that if v Bη and k N are such that j < < j ∈ g ∈ d ϕ Ada(v) ,e δ3 for all 0 j k,then Ada(v) Bη for all 0 j k. ³ ³ ´ ´ < ≤ ≤ ∈ ≤ ≤ JOURNAL OF MODERN DYNAMICS VOLUME 3, NO. 3 (2009), 457–477 ON A GENERALIZATIONOF LITTLEWOOD’S CONJECTURE 465

Let us describe how to obtain the lemma from these claims. Let λ be the minimum amongst the absolute values of the eigenvalues of Ada that are greater than 1. Choose δ δ3 as in Claim 2, M 0 as in Claim 1, and c c1 c2, where c1 = > = · g is as in Claim 1 and c satisfies d(ϕ(v ),ϕ(v )) c v v for any v , v B . 2 1 2 > 2k 1 − 2k 1 2 ∈ η Note that because of the choice of the norm on g, for v g one has ∈ j j j 0 j j j (2.8) Ad (v) Ad (v+) Ad (v ) Ad (v−) Ad (v+) λ v+ k a k=k a + a + a k≥k a k ≥ k k for any integer j . Let X ,Y and k N be as in the statement of the lemma. For i i ∈ any 0 j k we have: ≤ ≤ δ d(a j exp(X )exp(Y ), a j exp(X )exp(Y )) > 1 1 2 2 j j d(a exp(X )exp(Y )exp( Y )exp( X )a− ,e) = 1 1 − 2 − 2 j j d(a ϕ(u(X ,Y ))a− ,e) = i i j d(ϕ(Ad (u)),e) = a c Ad j (u) > 2k a k j c λ u+ ≥ 2 k k c c λj X X > 1 2 k 1 − 2k cλj X X . = k 1 − 2k We used the right-invariance of the metric in the first equality, the fact that δ δ1 1 < in the second, and the relation aϕ( )a− ϕ(Ad ( )) in the third. In the last set · = a · of inequalities, we used Claim 2 and the choice of c2 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. 1 NOTATION. If two positive numbers α and β satisfy r α β α for some r 0, < < r > we write α β. ∼r Proof of Claim 1. We use the notation of Lemma 2.3. Let0 ρ δ1 be such g 2 < g < that the map (v , v ) exp(v )exp( v ), takes B into expB . Since η was 1 2 7→ 1 − 2 ρ η chosen to be a bi-Lipschitz radius for the map exp,¡ ¢ there is a smooth function g 2 g w : B B which satisfies the relation ρ → η ¡ ¢ exp(w(v , v )) exp(v )exp( v ) for all v , v B g. 1 2 = 1 − 2 1 2 ∈ ρ u+ g Note that if v , v u+, then w(v , v ) B . Let X ,Y B ,i 1,2. The expres- 1 2 ∈ 1 2 ∈ η i i ∈ ρ = sions in (2.6) are equal to

(2.9) exp(w(X1, X2))exp(Adexp X2 w(Y1,Y2)). Let us sketch the line of reasoning we shall pursue: we show that w(v , v ) 1 2 ∼r v v for some r 0. We choose the constants carefully in such a way that k 1 − 2k > given X ,Y as in the statement of the claim, there exists a v g of length less i i ∈ than half of X X satisfying ϕ(v) exp(Ad w(Y ,Y )). It then follows k 1 − 2k = expX2 1 2 from (2.9) that

exp(u+) exp(w(X , X ))exp(v+) exp(w w(X , X ), v+ ). = 1 2 = 1 2 − ¡ ¢ JOURNAL OF MODERN DYNAMICS VOLUME 3, NO. 3 (2009), 457–477 466 URI SHAPIRA

Then, ignoring constants that will appear,

X1 X2 u+ w(X1, X2) v+ X1 X2 v+ k − k. k k=k + k>k − k−k k > 2 Let us turn now to the rigorous argument. The fact that η is a bi-Lipschitz radius g for exp implies the existence of a constant r 0 such that for any v , v B > 1 2 ∈ ρ (2.10) v v d(exp(v ),exp(v )) d(exp(v )exp( v ),e) w(v , v ) k 1 − 2k ∼r 1 2 = 1 − 2 ∼r k 1 2 k v v 2 w(v , v ) . ⇒k 1 − 2k ∼r k 1 2 k g Let M0 bound from above the operator norm of Adexpv as v ranges over Bρ. In (2.9), we have

M0 (2.11) AdexpX w(Y1,Y2) M0 w(Y1,Y2) Y1 Y2 . k 2 k ≤ k k < r 2 k − k 2 Let 0 δ2 ρ be such that 2δ2/r ρ. This implies by (2.10) that w(v1, v2) < < g < k g kg < ρ for all v1, v2 B . There exists some 0 ρ′ η such that exp(B ) ϕ(Bρ). ∈ δ2 < < ρ′ ⊂ g 1 Note that from the fact that exp is bi-Lipschitz on B and ϕ− is bi-Lipschitz on ρ′ g g exp B , it follows that there exist a constant r˜ such that for all w B , ³ ρ′ ´ ∈ ρ′ 1 (2.12) w ˜ ϕ− exp(w) . k k ∼r k k ¡ ¢ 2δ2 M0 2M0 g Let M max 2 , 4 . It follows from (2.11) that if Xi ,Yi B ,i 1,2 are = n r ρ′ r˜ r o ∈ δ2 = X1 X2 such that X u+ and Y Y k − k , then i ∈ k 1 − 2k < M M0 X1 X2 (2.13) AdexpX w(Y1,Y2) k − k, k 2 k < r 2M and by our choice of M

(2.14) Ad w(Y ,Y ) ρ′. k exp X2 1 2 k < g By the choice of ρ′, there exists some v B satisfying ∈ ρ (2.15) ϕ(v) exp(Ad w(Y ,Y )). = exp X2 1 2 Note that by (2.12) with w Ad w(Y ,Y ), (2.13), and the choice of M, = expX2 1 2 2 M0 X1 X2 r X1 X2 (2.16) v r˜ Adexp X w(Y1,Y2) v k − k k − k. k k ∼ k 2 k⇒k k < rr˜ 2M ≤ 2 The expressions in (2.6) andin(2.9) are equal to 0 0 (2.17) exp(u+)exp(u )exp(u−) exp(w(X , X ))exp(v+)exp(v )exp(v−). = 1 2 As noted above, the fact that Xi u+ implies that w(X1, X2) u+. From our ∈ u+ ∈ choice of ρ and the fact that v+, w(X , X ) B ,we getthat w w(X , X ), v+ 1 2 ∈ ρ 1 2 − ∈ u+ is deﬁned and satisﬁes ¡ ¢

(2.18) exp(w(X , X ))exp(v+) exp(w w(X , X ), v+ ). 1 2 = 1 2 − g ¡ g ¢ Because w( , ) takes values in B , and ϕ is injective on B , it follows from (2.17) · · η η and (2.18) that u+ w w(X , X ), v+ . = 1 2 − ¡ ¢ JOURNAL OF MODERN DYNAMICS VOLUME 3, NO. 3 (2009), 457–477 ON A GENERALIZATIONOF LITTLEWOOD’S CONJECTURE 467

Because of (2.10) and(2.16),

u+ w w(X , X ), v+ k k=k 1 2 − k 2 ¡ ¢ r w(X , X ) v+ > k 1 2 + k 2 r w(X , X ) v+ ≥ k 1 2 k−k k ¡ 4 ¢ 4 r r X1 X2 X1 X2 ≥ k − k − 2 k − k r 4 X1 X2 . = 2 k − k 4 Thus Claim 1 follows with the above choices of δ and M and with c r . 2 1 = 2 Proof of Claim 2. Let M Ad . Let0 δ δ satisfy 1 =k ak < 3 < 2 G g (2.19) Bδ ϕ Bη/M . 3 ⊂ ³ 1 ´ g Let v Bη and k N satisfy the assumptions of Claim 2. Assume by way of con- ∈ ∈ j g j 1 tradiction that there exists some 0 j k such that Ada(v) Bη but Ada+ (v) g ≤ < ∈ ∉ Bη. We conclude that j j g η M1 Ada(v) Ada(v) B . < k k⇒ ∉ η/M1 j This contradicts the assumption that ϕ Ad (v) BG and (2.19), because ϕ is a δ3 g ³ ´ ∈ injective on Bη. The proof of Lemma 2.3 is complete.

3. 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 [3] and the fundamental ideas appearing in [4]. The main hidden tool is the measure classiﬁcation theorem in [4]. What prevents us from citing known results is the fact that in the embedding τ: Yd , Xd 1 (2.1), the grids that are → + not Littlewood and thus have bounded Ad+ 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).

3.1. Beginning of proof of Theorem 1.4. Let Mi be an increasing sequence of compact sets in Xd 1 such that Xd 1 i Mi . Write Fi : y Yd : A+ τy Mi . + + = = ∈ d 1 ⊂ Lemma 2.2 implies that the set of exceptionsS to GLC satisﬁes© + ª (3.1) E F . d ⊂ i [i F is a compact subset of Y . We shall prove that dimF d 1 and conclude i d i ≤ − the proof of the theorem. Write L : τ(F ). As τ is, locally, a bi-Lipschitz map, it i = i will be enough to prove dimL d 1. As L τ(Y ), we wish to describe local i ≤ − i ⊂ d JOURNAL OF MODERN DYNAMICS VOLUME 3, NO. 3 (2009), 457–477 468 URI SHAPIRA neighborhoods in τ(Y ) in a convenient way. To do this, we take Ω A to be a d ⊂ d compact symmetric neighborhood of the identity, and we write (3.2) 0 ...... 0 ⋆ ⋆  . .   0 ... ⋆ .. .  .   . . . .    ......   ......     V1 :  . . . .  gd 1, V2 :  . gd 1. =   ∈ +  = . .. ⋆ ∈ +   . .  . .  ⋆ ... ⋆ .. .       0 ...... 0        0 ...... 0 0         V1 V2 Given δ 0 and x τ(Yd ) Xd 1, the set Ωexp(B )exp(B )x is a compact > ∈ ⊂ + δ δ neighborhood of x in τ(Yd ). As Li is compact, we can cover it by ﬁnitely many sets of the form L Ωexp(BV1 )exp(BV2 )x (for a suitable choice of points x L ). i ∩ δ δ ∈ i Now it is clear that for any x L , the intersection L Ωexp(BV1 )exp(BV2 )x is ∈ i i ∩ δ δ contained in

V1 V2 (3.3) Ω exp(B )exp(B )x z Xd 1 : A+ z ΩMi . · δ δ ∩ ∈ + d 1 ⊂ ³ © + ª´ Thus, the theorem will follow once we prove that

V1 V2 (3.4) dim exp(B )exp(B )x z Xd 1 : A+ z ΩMi 0, δ δ ∩ ∈ + d 1 ⊂ = ³ © + ª´ 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 [3]. 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 ⊂ Ad 1 and a compact set K Xd 1, we write + ⊂ + (3.5) KC : {x Xd 1 : C x K }. = ∈ + ⊂ Note that KC is a compact (possibly empty) C-invariant set.

LEMMA 3.1. Let C Ad 1 be a semigroup, a C, andK Xd 1 a compact set. If ⊂ + ∈ ⊂ + for some δ 0 and x K > ∈ u+(a) dim exp B x KC 0, ³ ³ δ ´ · ∩ ´ > then a acts with positive topological entropy on KC . For the proof of Theorem 1.4, we shall need the following generalization of Lemma 3.1.

LEMMA 3.2. Let C2 C1 Ad 1 be semigroups, ai Ci ,i 1,2, and K Xd 1 ⊂ ⊂ + ∈ = ⊂ + a compact set. Assume that there exists subspaces Vi of u+(ai ) such that for any b C , V u−(b). Then there exists δ 0 such that if for some x K ∈ 2 1 ⊂ > ∈ V1 V2 dim exp(B )exp(B ) x KC1 0 ³ δ δ · ∩ ´ > JOURNAL OF MODERN DYNAMICS VOLUME 3, NO. 3 (2009), 457–477 ON A GENERALIZATIONOF LITTLEWOOD’S CONJECTURE 469

then either a1 acts with positive topological entropy on KC1 or there exists a compact set K˜ K suchthat a acts with positive topological entropy on K˜ . ⊃ 2 C2 The following corollary goes along the lines of Proposition 4.1 from [3].

COROLLARY 3.3. Let C2 C1 Ad 1 be open cones, ai Ci ,i 1,2,andK Xd 1 ⊂ ⊂ + ∈ = ⊂ + a compact set. Assume that there exist subspaces Vi of u+(ai ) such that for any b C , V u−(b). Then there exists δ 0 such that for any x K ∈ 2 1 ⊂ > ∈

V1 V2 (3.6) dim exp(B )exp(B ) x KC1 0. ³ δ δ · ∩ ´ = Proof. In the proof of Proposition 4.1 from [3], it is shown that there cannot be an open cone C Ad 1 that acts on a compact invariant subset of Xd 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 nontrivial 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 [4].

3.2. 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 ,Ci ,Vi and ai , i 1,2, to ﬁt our needs in (3.4): we take V ,V to be as in (3.2) and = 1 2

C1 Ad+ 1, a1 diag(2,...,d 1,1/(d 1)!) , a2 diag(d 1,...,2,1/(d 1)!). = + = + + = + + Note that V u(a )+. Moreover, V u−(a ), thus we can choose C to be an 2 = 2 1 ⊂ 2 2 open cone containing a and contained in C such that for any b C ,V u−(b). 2 1 ∈ 2 1 ⊂ Finally, we take K ΩM . = i

Proof of Lemma 3.2. Note that from the fact that V u−(a ), it follows that the 1 ⊂ 2 sum V1 V2 is direct. Let V3 be any subspace of gd 1 such that gd 1 V1 V2 V3. + + + = ⊕ ⊕ Choose δ δ(K ) to be a bi-Lipschitz radius for K with respect to the above de- = composition (see §2.4 for notation). Assume also that δ satisﬁes the conclusion of Lemma 2.3 with a a . For the sake of brevity, we write B : BVi . Assume = 1 i = δ x K satisﬁes ∈ (3.7) dim exp(B )exp(B )x K 0. 1 2 ∩ C1 > ¡ ¢ Since δ is a bi-Lipschitz radius for K (with respect to the decomposition V V 1 ⊕ 2 ⊕ V3), (3.7) implies that the dimension of

F F (δ) (X ,Y ) B B : exp(X )exp(Y )x K = = ∈ 1 × 2 ∈ C1 © ª is positive. From the choice of the norm on gd 1 (see §2.4) andfrom the assump- + tion that for any b C2, V1 u−(b), it follows that Adb(X ) X for any X V1. ∈ ⊂ gd 1 k k≤k k ∈ Choose a compact set K˜ exp(B + )K . Denote by π the projection from B B ⊃ δ 2 1× 2 to B2. There are two cases:

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Case 1. Assume dimπ (F ) 0. We claim that exp(π (F )) x K˜ . To see this, 2 > 2 ⊂ C2 note that if Y π (F ), then there exists some X B such that exp(X )exp(Y )x ∈ 2 ∈ 1 ∈ K , andso for any b C we have C1 ∈ 2 gd 1 b exp(Y )x exp(Ad ( X )b)exp(X )exp(Y )x exp(B + )K K˜. = b − ∈ δ ⊂ ˜ Now exp(π2(F )) x exp(B2) x KC2 and therefore, positivity of the dimension ⊂ · ∩ ˜ of π2(F ) implies the positivity of the dimension of exp(B2) x KC2 . We apply · ∩ ˜ Lemma 3.1 and conclude that a2 acts with positive topological entropy on KC2 . Case 2. Assume dimπ (F ) 0 and let us write dimF 3ρ with ρ 0. We will 2 = = > show that a1 acts with positive topological entropy on KC1 . Recall the notation of Lemma 2.3 (applied to a1). We shall find for arbitrarily large integers n finite sets S F with the following properties: n ⊂ if (X ,Y ) S ,i 1,2 are distinct, then • i i ∈ n = n n (3.8) X X λ− , Y Y λ− /M, k 1 − 2k > k 1 − 2k < and ρ nρ S M − λ . • | n| > Given two distinct points in S , (X ,Y ),i 1,2, let us analyze the rate at which n i i = exp(Xi )exp(Yi )x drift apart from each other under the action of powers of a1. j For any j 0, we have that a exp(X )exp(Y )x K by the definition of F , and so, ≥ 1 i i ∈ if the distance between these two points is less than δ (which is also an isometry radius for K ), we have

j j (3.9) d(a1 exp(X1)exp(Y1)x, a1 exp(X2)exp(Y2)x) d(a j exp(X )exp(Y ), a j exp(X )exp(Y )). = 1 1 1 1 2 2 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 j j k k n d(a exp(X )exp(Y )x, a exp(X )exp(Y )) cλ X X cλ − . 1 1 1 1 2 2 ≥ k 1 − 2k > In particular, if we set ǫ min{c,δ}, then we must have some 0 j n for which 0 = ≤ ≤ j j d(a exp(X )exp(Y )x, a exp(X )exp(Y )x) ǫ . 1 1 1 1 2 2 > 0 Thus, exp(X )exp(Y )x : (X ,Y ) S is an (n,ǫ )-separated set for (K , a ). From ∈ n 0 C1 1 here, it© is easy to derive the positivityª of the entropy by the bound we have on the size of Sn:

1 1 ρ nρ htop (KC , a1) limsup log Sn lim log(M − λ ) ρ logλ 0. 1 ≥ n | | ≥ n n = > To build the sets Sn with the above properties for arbitrarily large n’s, we argue as follows. By deﬁnition of the dimension, one can ﬁnd a sequence ǫ 0 such k ց that S (F ) (1/ǫ )2ρ. ǫk > k n n 1 Choose n such that λ− k ǫ λ− k + . It follows that k ր ∞ ≤ k < 2nk ρ 2ρ (3.10) S nk (F ) S (F ) λ − . λ− ≥ ǫk > JOURNAL OF MODERN DYNAMICS VOLUME 3, NO. 3 (2009), 457–477 ON A GENERALIZATIONOF LITTLEWOOD’S CONJECTURE 471

On the other hand, because we assume dimπ (F ) 0, for any large enough n, 2 = log(N n (π2(F ))) λ− /M ρ. log(λn M) < Hence nρ ρ (3.11) N n (π (F )) λ M . λ− /M 2 < (k) N n Write Nk : λ− k /M (π2(F )) and let Ei ,i 1... Nk be a covering of π2(F ) by = n = n ρ ρ subsets of B of diameter less than λ− k /M. Since N λ k M , by (3.10) and 2 k < by the pigeonhole principle, there must exist some 1 i N with ≤ k ≤ k 1 (k) n ρ ρ S n k (3.12) λ− k (π2− (E ) F ) λ M − . ik ∩ >

nk 1 (k) Deﬁne Sn to be a maximal λ− -separated set in π− (E ) F . By construction, k 2 ik ∩ Snk has the desired properties.

Proof of Corollary 1.5. The projection π: Y X cannot increase dimension, d → d and therefore π(Ed ), the set of lattices which are not grid-Littlewood, is a countable union of sets of upper box dimension d 1. d ≤ − Denote by p : Xd R Yd the map p(x, v) x v. It is bi-Lipschitz with a × → = d + d countable ﬁber, and so if we denote by p2 : Xd R R the natural projection, 1 × → then p2(p− (Ed )), 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 1 dim π− (x) E 0. ∩ d > It follows that if Ω A is a compact¡ neighborhood¢ of the identity, then ⊂ d 1 Ω π− (x) E ∩ d ¡ ¢ has dimension greater than dim A d 1, which is a contradiction to Theorem d = − 1.4.

Proof of Corollary 1.7. Positivity of the dimension of a subset L X , transverse ⊂ d to the Ad -orbits, means that Ad L contains a compact set of dimension greater then dim A d 1. By Theorem 1.4, such a set must contain a grid-Littlewood d = − lattice. If the dimension of the A -orbit closure of a lattice x X is greater then d ∈ d 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 asserted in the statement. It shows that the set

y Y : A+ τ is bounded ∈ d d 1 y © + ª is a countable union of compact sets of upper box dimension d 1. The corol- ≤ − laries of Theorem 1.4 have corresponding versions as well.

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4. PROOFS OF THEOREMS 1.8 AND 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 ∈ and (α,1)t in the plane is grid-Littlewood, and moreover that for any grid y 1 ∈ π− (x) one has that A3+τy is unbounded in X3. We now untie the definitions and translate this information to Diophantine information on α. t 2 Given any vector v (β,γ) R , if we write y x v, the fact that A+τ is = ∈ = + 3 y unbounded is equivalent (in light of Mahler’s compactness criterion) to the existence of a sequence a A+ going to infinity (i.e., leaving any compact subset) i ∈ 3 and a sequence of vectors 0 w τ , such that a w 0 in R3. Write 6= i ∈ y i i → t s (t s ) t 3 a : diag(e i ,e i ,e− i + i ); w (k ℓ α n β, ℓ n γ, n ) R , i = i = i + i + i i + i i ∈ where k ,ℓ ,n Z and are not all equal to zero for each i. Then i i i ∈ t s (t s ) (4.1) max e i k ℓ α n β ;e i ℓ n γ ;e− i + i n 0. i + i + i i + i | i | → © ¯ ¯ ¯ ¯ ª As t , s 0, it follows¯ from (4.1) first¯ that¯ℓ (n γ¯ )∗, and then that i i ≥ i = i (4.2) max n γ ; (n γ)∗α n β 0. 〈 i 〉 〈 i + i 〉 → © ª Taking the product of the three quantities in (4.1), we get

(4.3) n n γ (n γ)∗α n β 0. | i |〈 i 〉〈 i + i 〉→ We are left to justify why we can choose the n ’s such that n . Note that i | i | → ∞ if γ is rational and 1,α,β are linearly dependent over Q, then the statement of the theorem is clear. Assume that n is bounded. We might as well assume that | i | it is constant. Then ℓ is also constant, and moreover n γ ℓ , so γ is ratio- i i = − i nal. 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, 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 , if we write ∈ 2 y x v Y , then A+τ is unbounded in X . Given α R, we apply this to the = + ∈ 2 3 y 3 ∈ lattice spanned by (1,0)t and (α,1)t and continue as in the proof of Theorem 1.8 above.

5. LATTICES THAT SATISFY GLC In this section, we shall explicitly build grid-Littlewood lattices in Rd for d 3. ≥ In fact, these lattices will possess a much stronger property.

DEFINITION 5.1. A grid y Y is Littlewood of finite type if there exists a nonzero ∈ d integer n such that N(ny) 0. A lattice x Xd is grid-Littlewood of finite type if 1 = ∈ any y π− (x) is Littlewood of finite type. ∈ DEFINITION 5.2. A grid y Yd is rational if y is an element of finite order in the 1 d ∈ group π− (π(y)) R /π(y). = The following list of observations is left to be verified by the reader.

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PROPOSITION 5.3.

1. The set of grids that are Littlewoodof finite type is Ad invariant. 2. If y, y Y , y A y,and y is Littlewoodof finitetype, then y isLittlewood 0 ∈ d 0 ∈ d 0 of finite type. 3. If x, x X , x A x, and x is grid-Littlewood of finite type, then x is 0 ∈ d 0 ∈ d 0 grid-Littlewood of finite type. 4. Any rational grid is Littlewood of finite type. 5. If x X is grid-Littlewood of finite type and x X is any lattice, then 1 ∈ d1 2 ∈ d2 x1 x2 Xd d is grid-Littlewood of finite type. ⊕ ∈ 1+ 2 6. If x , x X are such that x is grid-Littlewood of finite type and there ex- 1 2 ∈ d 1 ists some c 0 such that cx is commensurable with x , then x is grid- > 1 2 2 Littlewood of finite type. 7. The standard lattice Zd is not grid-Littlewood of finite type. In fact, Zd + v is not Littlewood of finite type for any vector v Rd with all coordinates ∈ irrational.

For x Xd , denote by Ad,x its stabilizer in Ad . Note that Ad,x acts on the torus 1 ∈ π− (x) as a group of automorphisms. From (2) and (4) of Proposition 5.3, we deduce the following. 1 1 LEMMA 5.4. If for any grid y π− (x), A y π− (x) contains a grid that is Lit- ∈ d,x ⊂ tlewood of finite type, then x is grid-Littlewood of finite type. In particular, if for 1 any grid y π− (x), A y contains a rational grid, then x is grid-Littlewood of ∈ d,x finite type. 1 Recall that a group of automorphisms of a torus π− (x) (x X ) is called ID ∈ d if any infinite invariant set is dense. The following is a weak version of Theorem 2.1 from [1].

THEOREM 5.5 (Theorem 2.1 [1]). If the stabilizer A of a lattice x X under d,x ∈ d theactionof Ad satisfies 1. there exists some a A such that for any n, the characteristic polynomial ∈ d,x of an (which is necessarily over Q) is irreducible, 2. foreach 1 i d, there exists a diag(a ... a ) A with a 1, and ≤ ≤ = 1 d ∈ d,x i 6= 3. there exist a , a A which are multiplicatively independent (that is, if 1 2 ∈ d,x ak am 1, then k m 0), 1 2 = = = then Ad,x is ID. Wenow turn to the construction of a family of lattices that are grid-Littlewood of finite type. Let K be a totally real number field of degree d over Q. The ring of integers of K will be denoted by OK .

DEFINITION 5.6. 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

OΛ {x K : xΛ Λ}. = ∈ ⊂ JOURNAL OF MODERN DYNAMICS VOLUME 3, NO. 3 (2009), 457–477 474 URI SHAPIRA

It can be easily veriﬁed that OΛ is a ring for any lattice Λ in K . Moreover, the units in this ring are exactly O∗ {ω K : ωΛ Λ}. Dirichlet’s unit theorem Λ = ∈ = states the following.

THEOREM 5.7 (Dirichlet’s unit theorem). For any lattice Λ inK,thegroupofunits d 1 O∗ is isomorphic to { 1} Z − . Λ ± × Let σ1 ...σd be an ordering of the different embeddings of K into the reals. Define ϕ: K Rd to be the map whose i’th coordinate is σ . If we endow Rd → i 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 Rd ). It is well-known that if Λ is a lattice in K , then ϕ(Λ) d is a lattice in R . Let us denote by xΛ the point in Xd obtained by normalizing the covolume of ϕ(Λ)tobe1. Werefertosuch alatticeasa lattice coming from a number field. Because ϕ is a homomorphism, d ϕ(OΛ∗ ) a R : axΛ xΛ . ⊂ n ∈ = o We can identify the linear map obtained by left multiplication by a Rd on Rd ∈ 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∗ ) is a subgroup of the ± Λ stabilizer of xΛ 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 SLd , we replace OΛ∗ by the subgroup OΛ∗, of totally positive units (that is, units for which all embeddings + are positive ). It is a subgroup of finite index in OΛ∗ . We conclude that ϕ will map OΛ∗, into Ad,xΛ (using our identification of vectors and diagonal matrices). + LEMMA 5.8. If xΛ Xd is a lattice coming from a totally real number field K of ∈ 1 degree d 3,then A Λ is an ID group of automorphisms of π− (xΛ). ≥ d,x Proof. It is enough to check that conditions (1),(2), and (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 ≥ n follows. We will show that there exists α O∗ such that for any n, α generates ∈ K K (this is enough because OΛ∗, is of finite index in OK∗ ). Let F1,...,Fk be a list of all the subfields of K . If for a subset+ B K we write ⊂ pB : x K : n such that xn B , = ∈ ∃ ∈ © ª then we need to show that k (5.1) O∗ O∗ . K à Fi 6= ; [1 q Fix a proper subfield F of K . Note that the following is an inclusion of groups

OF∗ OF∗ OK∗ . Thus, Dirichlet’s unit theorem will imply (5.1) once we prove ⊂ q ⊂ that OF∗ is of ﬁnite index in OF∗. We shall give a bound on the order of elements q JOURNAL OF MODERN DYNAMICS VOLUME 3, NO. 3 (2009), 457–477 ON A GENERALIZATIONOF LITTLEWOOD’S CONJECTURE 475 in the quotient OF∗ OF∗, thusshowing thatthe groups are of the same rank. It is q enough to show that± there exists some integer n such that if x K satisﬁes xn 0 ∈ ∈ F for some n, then xn0 F . Let x K be such an element. Denote by σ ...σ the ∈ ∈ 1 r different embeddings of F intothereals, andfor 1 i r , denote by σ , j 1... s ≤ ≤ ij = the different extensions of σ to an embedding of K into the reals. Thus d rs i = and σ are all the different embeddings of K into the reals. Note that xn F if ij ∈ n n σi j (x) n and only if σi1(x ) σis(x ) for any1 i r , i.e., if and only if ( ) 1 =···= ≤ ≤ σik (x) = 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 the next theorem.

THEOREM 5.9. Any lattice coming from a totally real number ﬁeld of degree d 3 ≥ is grid-Littlewood of ﬁnite type.

Proof. Let xΛ X be a lattice coming from a totally real number field of degree ∈ d d 3. Using Lemma 5.8 and Lemma 5.4, we see that the theorem will follow if we ≥ 1 show that any finite Ad,xΛ -invariant set in π− (xΛ) contains only rational grids. 1 Assume that y π− (xΛ) lies in a finite invariant set. It follows that there exists ∈ e a ϕ(OΛ∗, ) with 6= ∈ + (5.2) ay y. = Write xΛ cϕ(Λ), y xΛ v, and a ϕ(ω). Then from (5.2), it follows that there = = + = exists θ Λ such that in the algebra Rd , ∈ 1 v(ϕ(ω) 1) cϕ(θ) v cϕ(θ(ω 1)− ). − = ⇒ = − Since K is spanned over Q by Λ, we see that v is in the Q-span of cϕ(Λ) xΛ, and = hence y is a rational grid as desired.

As a corollary of the ergodicity of the Ad action on Xd and part (3) of Proposi- tion 5.3, we get the following (we refer the reader to [8] for a stronger result).

COROLLARY 5.10. Almost any x X is grid-Littlewood of ﬁnite type for d 3. ∈ d ≥ The following result appears for example in [5].

THEOREM 5.11. The compact orbits for Ad in Xd 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 part (3) of Proposition 5.3, combined with Theorem 5.9. We state it separately because of its interesting resemblance to Theorem 1.3 from [5].

COROLLARY 5.12. For x X (d 3), if A x containsa compact A -orbit, then x ∈ d ≥ d d is grid-Littlewood of ﬁnite type. The reader is referred to [8], where a signiﬁcant strengthening of this corollary is established. Let us end this paper with two open problems which emerge from our discussion.

JOURNAL OF MODERN DYNAMICS VOLUME 3, NO. 3 (2009), 457–477 476 URI SHAPIRA

PROBLEM 5.13. Give an explicit example of a Littlewood lattice in dimension 2. In particular,prove that any lattice with a compact A2-orbit is Littlewood.

PROBLEM 5.14. Do there exist two-dimensional lattices that are grid-Littlewood of ﬁnite type?

6. APPENDIX Proof of Lemma 3.1. Let notation be as in Lemma 3.1. The statement of Lemma 2.3 simpliﬁes when one chooses the Yi ’s to be zero in the original statement:

LEMMA (Lemma 2.3, simplified version). For a fixed element e a Ad 1, there 6= ∈ + exist λ 1 and η,c 0 such that for any X B u+(a),i 1,2, if for an integerk, for > > i ∈ η = 0 j k ≤ ≤ d(a j exp(X ), a j exp(X )) η, 1 2 < then for any 0 j k ≤ ≤ d(a j exp(X ), a j exp(X )) cλj X X . 1 2 ≥ k 1 − 2k We apply this lemma for the element a Ad 1 appearing in the statement of ∈ + Lemma 3.1. Let0 δ′ max η,δ be a bi-Lipschitz radius for K with respect to < < the chart exp (see §2.4 for notation).© ª Cover the compact set exp(B u+(a))x K by δ ∩ C u+(a) finitely many sets of the form exp(B )yi KC , for a suitable choice of points δ′ ∩ u+(a) yi KC . By assumption, there exists an i such that dim exp(B )yi KC 0. ∈ ³ δ′ ∩ ´ > Because δ′ is a bi-Lipschitz radius, the dimension of

u+(a) (6.1) F F (δ′) : X B : exp(X )yi KC = = n ∈ δ′ ∈ o is positive. Denote it by 2ρ. By the deﬁnition of dimension, this means that there ρ − exists a sequence ǫk 0, and ǫk separated sets Sk F , such that Sk ǫk . Let ց − n n⊂ 1 | | > n be a sequence such that λ− k ǫ λ− k + . Let X , X S be two k ր ∞ ≤ k < 1 2 ∈ k distinct points. Because δ′ is also an isometry radius for K , if ℓ is an integer j j such that d(a exp(X )y , a exp(X )y ) δ′ for all 0 j ℓ, then the simpliﬁed 1 i 2 i < ≤ ≤ version of Lemma 2.3 stated above implies that for all 0 j ℓ ≤ ≤ j j δ′ d a exp(X1)yi , a exp(X2)yi > ³ ´ j j (6.2) d a exp(X1), a exp(X2) = ³ ´ j j j n cλ X X cλ ǫ cλ − k . ≥ k 1 − 2k ≥ k > This means that if ǫ min δ′,c , exp(X )y : X S is an (ǫ ,n ) separating 0 = i ∈ k 0 k − set for the action of a on KC .© We concludeª © that ª

1 ρ logǫk htop(KC , a) lim log Sk lim − ρ logλ 0. ≥ k nk | | ≥ k nk ≥ > Thus we achieve the desired conclusion.

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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 Ein- siedler, for their warm hospitality during a visit in which much of the research was conducted.

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[1] D. Berend, Multi-invariant sets on tori, Trans. A.M.S., 280 (1983), 509–532. [2] J. W. S. Cassels and H. P. F. Swinnerton-Dyer, On the product of three homogeneous linear forms and the indeﬁnite ternary quadratic forms, Philos. Trans. Roy. Soc. London. Ser. A., 248 (1955), 73–96. [3] M. Einsiedler and D. Kleinbock, Measure rigidity and p-adic Littlewood-type problems, Com- pos. Math., 143 (2007), 689–702. [4] M. Einsiedler, A. Katok and E. Lindenstrauss, Invariant measures and the set of exceptions to Littlewood’s conjecture, Ann. of Math., 164 (2006), 513–560. [5] E. Lindenstrauss and B. Weiss, On sets invariant under the action of the diagonal group, Er- godic Theory Dynam. Systems, 21 (2001), 1481–1500. [6] G. A. Margulis, Oppenheim conjecture, Fields Medallists’ Lectures lectures, World Sci. Ser. 20th Century Math., vol. 5, 272–327, World Sci. Publishing, River Edge, NJ, 1997. [7] G. A. Margulis, Problems and conjectures in rigidity theory, Mathematics: frontiers and per- spectives, 161–174, Amer. Math. Soc., Providence, RI, 2000. [8] U. Shapira, A solution to a problem of Cassels and Diophantine properties of cubic numbers, Preprint. arXiv:0810.4289

URI SHAPIRA : Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel

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