Solvability of Rado systems in D-sets

Radoʹs Theorem characterizes the systems of homogeneous linear equations having the property that for any finite partition of the positive integers one cell contains a solution to these equations. Furstenberg and Weiss proved that solutions to those systems can in fact be found in every central set. (Since one cell of any finite partition is central, this generalizes Radoʹs Theorem.) We show that the same holds true for the larger class of D-sets. Moreover we will see that the conclusion of Furstenbergʹs Central Sets Theorem is true for all sets in this class.