On the structure of sets of large doubling

We investigate the structure of finite sets A ⊆ Z where | A + A | 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 ∘ [ g ] sets, and enables us to construct a Λ ( 4 ) set which does not contain large B 2 [ g ] or B 2 ∘ [ g ] sets.