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 to
Diophantine 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.
Mathematics Subject Classification:
Primary: 37A17, 37N99; Secondary: 11H46, 11J20, 11J25, 11J3.
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