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

作者：Shapira Uri^*

来源：Annals of Mathematics, 2011, 173(1): 543-557.

DOI：10.4007/annals.2011.173.1.11

We prove that almost any pair of real numbers alpha, beta, satisfies the following inhomogeneous uniform version of Littlewood's conjecture: (C1) for all gamma, delta is an element of R, liminf(vertical bar n vertical bar -> infinity)vertical bar n vertical bar < n alpha - gamma > < n beta - delta > = 0, where <.> 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 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.

• 出版日期2011-1