On a generalization of Littlewood's conjecture

TL;DR

This paper introduces GL-lattices in R^d, conjectures their universality, and proves their existence in higher dimensions using dynamical systems and Diophantine approximation techniques.

Contribution

It defines GL-lattices, formulates the GLC conjecture that all lattices are GL, and provides explicit constructions and dimension bounds related to this conjecture.

Findings

Existence of GL lattices in dimensions d >= 3.

Dimension bounds for exceptions to the GLC.

Connections between GLC and Margulis's conjecture on bounded orbits.

Abstract

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.