  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.
