A solution to an open problem of Cassels and Diophantine properties of cubic numbers. The abstract is:

Abstract: We prove existence of real numbers x,y, possessing the following property:

For any real a,b, liminf |n| ||nx - a|| ||ny - b|| = 0, where ||c|| denotes the distance of c to the nearest integer.

This answers a 50 years old question of Cassels. The most interesting part of the result is that there are algebraic numbers with the above property!