A solution to a problem of Cassels and Diophantine properties of cubic numbers


We prove that almost any pair of real numbers $\alpha,\beta$, satisfies the following inhomogeneous uniform version of Littlewood’s conjecture: $$\begin{align}\label{C1abst}\tag{C1} \forall \gamma,\delta\in\mathbb{R},\quad \liminf_{|n|\to\infty} \left|n\right|\langle n\alpha-\gamma \rangle\langle n\beta-\delta\rangle=0, \end{align}$$ where $\langle\cdot\rangle$ denotes the distance from the nearest integer. The existence of even a single pair that satisfies statement (C1), solves a problem of Cassels from the 50’s. We then prove that if $1,\alpha,\beta$ span a totally real cubic number field, then $\alpha,\beta$, satisfy (C1). This generalizes a result of Cassels and Swinnerton-Dyer, which says that such pairs satisfy Littlewood’s conjecture. It is further shown that if $\alpha,\beta$ are any two real numbers, such that $1,\alpha,\beta$, are linearly dependent over $\mathbb{Q}$, they cannot satisfy (C1). The results are then applied to give examples of irregular orbit closures of the diagonal group of a new type. The results are derived from rigidity results concerning hyperbolic actions of higher rank commutative groups on homogeneous spaces.