A solution to a problem of Cassels and Diophantine properties of cubic numbers
A solution to a problem of Cassels and Diophantine properties of cubic numbers
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立方数卡塞尔和丢番图性质问题的一个解
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发表时间：
2008
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通讯作者：
Uri Shapira
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作者：
Uri Shapira
我们证明了几乎任何一对真实的数满足下列非齐次一致版本的Littlewood猜想：8; 2 R; lim inf jnj！其中hi表示与最近整数的距离。满足陈述（C1）的甚至一对的存在，解决了50年代的卡塞尔问题。证明了如果1;;跨越一个全真实的立方数eld，则满足（C1）。这推广了Cassels和Swinnerton-Dyer的一个结果，即这样的对满足Littlewood猜想。进一步证明了：如果;是任意两个真实的数，使得1;;，对Q线性相关，则它们不能满足（C1）.然后，应用结果给出了一种新类型的对角群的不规则轨道闭包的例子。这些结果是从齐型空间上高阶交换群的双曲作用的刚性结果导出的。
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.