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*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.

We show that in positive characteristic the homogeneous probability measure supported on a periodic orbit of the diagonal group in the space of $2$-lattices, when varied along rays of Hecke trees, may behave in sharp contrast to the zero characteristic analogue: For a large set of rays, the measures fail to converge to the uniform probability measure on the space of $2$-lattices. More precisely, we prove that when the ray is rational there is uniform escape of mass, that there are uncountably many rays giving rise to escape of mass, and that there are rays along which the measures accumulate on measures which are not absolutely continuous with respect to the uniform measure on the space of $2$-lattices.

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