The Gap Structure of a Family of Integer Subsets.

In this paper we investigate the gap structure of a certain family of subsets of N which produces counterexamples both to the "density version" and the "canonical version" of Brown's lemma. This family includes the members of all complementing pairs of N. We will also relate the asymptotical gap structure of subsets of N with their density and investigate the asymptotical gap structure of monochromatic and rainbow sets with respect to arbitrary infinite colorings of N.