Solvability of Rado systems in D-sets

TL;DR

This paper extends the known results about the solvability of Rado systems from central sets to the larger class of D-sets, showing solutions exist in all D-sets and that Furstenberg's Central Sets Theorem applies to them.

Contribution

It generalizes the existence of solutions to Rado systems from central sets to D-sets and confirms the applicability of Furstenberg's Central Sets Theorem to D-sets.

Findings

Solutions to Rado systems exist in all D-sets.

Furstenberg's Central Sets Theorem applies to D-sets.

D-sets form a larger class than central sets for these properties.

Abstract

Rado's Theorem characterizes the systems of homogenous 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 Rado's Theorem.) We show that the same holds true for the larger class of $D$-sets. Moreover we will see that the conclusion of Furstenberg's Central Sets Theorem is true for all sets in this class.