On the Structure of Sets of Large Doubling

Abstract

We investigate the structure of finite sets $A\subseteq \mathbb{Z}$ 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 $B_2^{\circ }[g]$ sets, and enables us to construct a $\Lambda (4)$ set which does not contain large $B_2[g]$ or $B_2^{\circ }[g]$ sets.

23 pages, changed title, revised version reflects work of Meyer that we were previously unaware of