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

Details 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

2008-10-23

arxiv.org/abs/0810.4289v2

Mathematics

Dynamical Systems

12 pages

Scientific paper

We prove that almost any pair of real numbers a,b, satisfies the following inhomogeneous uniform version of Littlewood's conjecture: (*) forall x,y in R, liminf_{|n|\to\infty} |n| = 0, where <-> denotes the distance from the nearest integer. The existence of even a single pair that satisfies (*), solves a problem of Cassels from the 50's. We then prove that if 1,a,b span a totally real number field, then a,b, satisfy (*). It is further shown that if 1,a,b, are linearly dependent over Q, a,b cannot satisfy (*). The results are then applied to give examples of irregular orbit closures of the diagonal groups of a new type.

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