On a generalization of Littlewood's conjecture

2008/10/23 by Uri Shapira, Shapira, Uri
Mathematics · #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #math.DS #math.NT

paper · pdf · doi:10.48550/arxiv.0810.4298

17 pages

arxiv created 2009/05/07 · arxiv updated 2009/12/01

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 Z² 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.

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