Journal of Modern Dynamics (JMD)

On a generalization of Littlewood's conjecture

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

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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.