A Solution to a Problem of Cassels and Diophantine Properties of Cubics

We prove that almost any pair of real numbers α, β, satisfies the following inhomogeneous uniform version of Littlewood’s conjecture: For any real γ, δ, 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 an open 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). In particular, pairs of real numbers from the same quadratic field, can never satisfy (0.1). The results are derived from rigidity results concerning hyperbolic actions of higher rank commutative groups on homogeneous spaces.