Multiplicatively badly approximable vectors

The Littlewood Conjecture states that for all pairs of real numbers (α, β) the product  |q||qα + p1||qβ + p2|

becomes arbitrarily close to 0 when the vector (q,p1,p2) ranges in Z^3 and q ≠ 0. To date, despite much progress, it is not known whether this statement is true. In this talk, I will discuss a partial converse of the Littlewood Conjecture, where the factor |q| is replaced by an increasing function f(|q|). More specifically, following up on the work of Badziahin and Velani, I will be interested in determining functions f for which the above product and its higher dimensional generalizations stay bounded away from 0 for at least one pair (α, β) ∈ R^2. I will show how this problem can be reduced to counting lattice points in certain distorted boxes, which will, in turn, require careful estimates of the minimum of the lattice and of its dual.