Abstract
We present a class of lattices in $\R^d$ ($d\ge 2$) 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 $\Z^2$ is grid-Littlewood. We then prove the existence of grid-Littlewood lattices by first establishing a dimension bound for the set of possible exceptions. The existence of vectors (grid-Littlewood-vectors) in $\R^d$ with special Diophantine properties is proved by similar methods. Applications toDiophantine approximations are given. For dimension $d\ge 3$, 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.
Highlights
We present a class of lattices in Rd (d ≥ 2) which we call GL − lattices and conjecture that any lattice is such
We identify Xd with the homogeneous space Gd/Γd in the following manner: For g ∈ Gd, the coset gΓd represents the lattice spanned by the columns of g
Let us sketch the line of proof we shall pursue: We show that w(v1, v2) ∼r ||v1 − v2||, for some r > 0
Summary
We define the following metric on X ( denoted by d(·, ·)) Under these metrics, for any compact set K ⊂ X there exist an isometry radius ǫ(K), i.e. a positive number ǫ such that for any x ∈ K, the map g → gx is an isometry between. We choose the constants carefully in such a way that given Xi, Yi as in the statement of the claim, there exist a v ∈ g of length less than half of ||X1 − X2||, with φ(v) = exp Adexp X2w(Y1, Y2) It follows from (2.10) that exp u+ = exp w(X1, X2) exp v+ = exp w w(X1, X2), −v+. This contradicts the assumption that φ (Adja(v)) ∈ BδG3 and (2.20) because φ is injective on Bηg
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