On a generalization of Littlewood's conjecture

Pages: 457 - 477,
Issue 3,
July
2009 doi:10.3934/jmd.2009.3.457

Uri Shapira - Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel (email)

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 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.

Keywords: Littlewood's conjecture, diagonal flow, inhomogeneous lattices, product of inhomogeneous linear forms.

Mathematics Subject Classification: Primary: 37A17, 37N99; Secondary: 11H46, 11J20, 11J25, 11J37.

Received: April 2009;
Revised:
June 2009;
Available Online: August 2009.