A Solution to a Problem of Cassels and Diophantine Properties of Cubic Numbers

We prove that almost any pair of real numbers α, β, satisfies the following inhomogeneous uniform version of Littlewood’s conjecture: ∀γ, δ ∈ R, lim inf |n|→∞ |n| 〈nα− γ〉〈nβ − δ〉 = 0, (0.1) where 〈·〉 denotes the distance from the nearest integer. The existence of even a single pair that satisfies (0.1), solves a problem of Cassels [Ca] from the 50’s. We then prove that if 1, α, β span a totally real cubic number field, then α, β, satisfy (0.1). This generalizes a result of Cassels and Swinnerton-Dyer, which says that such pairs satisfies Littlewood’s conjecture. It is further shown that if α, β are any two real numbers, such that 1, α, β, are linearly dependent over Q, they cannot satisfy (0.1). The results are then applied to give examples of irregular orbit closures of the diagonal groups of a new type. The results are derived from rigidity results concerning hyperbolic actions of higher rank commutative groups on homogeneous spaces.