Description
An asymmetric version of the Littlewood conjecture
In this talk, we study an asymmetric version of the Littlewood
conjecture proposed by Y. Bugeaud. A parameter σ ∈ [0,1] being fixed, we study the set B(σ) of those pairs of real numbers (x,y) such thatinfq≥1(q · max(∥qx∥ ∥qy∥)1+σ min(∥qx∥ ∥qy∥)1−σ ) > 0. Counter-examples to the Littlewood conjecture would belong to B(0) and appear as an interpolation from the set B(1) corresponding to the badly approximable vectors in dimension 2. We prove that for every σ ∈ [0,1], B(σ) has Hausdorff dimension 2, and propose some natural conjectures around such sets. Joint work in progress with F. Adiceam.
