A solution to a problem of Cassels and Diophantine properties of cubic numbers
Abstract
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|∞ |n|<na - x> <nb - y> = 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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