A solution to a problem of Cassels and Diophantine properties of cubic numbers
Abstract
Description
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|<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.
12 pages
12 pages