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

Abstract

We prove that almost any pair of real numbers ; , satises the following inhomogeneous uniform version of Littlewood’s conjecture: 8; 2 R; lim inf jnj!1 jnjhn ihn i = 0; (C1) wherehi denotes the distance from the nearest integer. The existence of even a single pair that satises statement (C1), solves a problem of Cassels from the 50’s. We then prove that if 1;; span a totally real cubic number eld, then ; , 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 ; are any two real numbers, such that 1;; , are linearly dependent over 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.