On the structure of sets of large doubling

We investigate the structure of finite sets A subset of Z where vertical bar A + A vertical bar is large. We present a combinatorial construction that serves as a counterexample to natural conjectures in the pursuit of an "anti-Freiman" theory in additive combinatorics. In particular, we answer a question along these lines posed by O'Bryant. Our construction also answers several questions about the nature of finite unions of B(2)[g] and B(2)(o)[g] sets, and enables us to construct a Lambda(4) set which does not contain large B(2)[g] or B(2)(o)[g] sets. (C) 2011 Elsevier Ltd. All rights reserved.