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July  2009, 3(3): 457-477. doi: 10.3934/jmd.2009.3.457

On a generalization of Littlewood's conjecture

Uri Shapira 1,

1. 

Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel

Received  April 2009 Revised  June 2009 Published  August 2009

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: diagonal flow, Littlewood's conjecture, inhomogeneous lattices, product of inhomogeneous linear forms..
Mathematics Subject Classification: Primary: 37A17, 37N99; Secondary: 11H46, 11J20, 11J25, 11J3.
Citation: Uri Shapira. On a generalization of Littlewood's conjecture. Journal of Modern Dynamics, 2009, 3 (3) : 457-477. doi: 10.3934/jmd.2009.3.457
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