On the structure of sets of large doubling

We investigate the structure of finite sets A⊆ZA⊆Z where |A+A||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 B2[g]B2[g] and B2∘[g] sets, and enables us to construct a Λ(4)Λ(4) set which does not contain large B2[g]B2[g] or B2∘[g] sets.