On the Structure of Sets of Large Doubling

TL;DR

This paper constructs a combinatorial example of a finite set with large sumset size, challenging existing conjectures in additive combinatorics and revealing new structural insights.

Contribution

It provides a counterexample to natural conjectures and answers open questions about the structure of sets with large doubling and their unions.

Findings

Counterexample to anti-Freiman conjectures

Answers to questions on unions of $B_2[g]$ and $B^ullet_2[g]$ sets

Construction of a $ ext{Lambda}(4)$ set lacking large $B_2[g]$ or $B^ullet_2[g]$ subsets

Abstract

We investigate the structure of finite sets $A \subseteq \Z$ 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 $B^\circ_2[g]$ sets, and enables us to construct a $\Lambda(4)$ set which does not contain large $B_2[g]$ or $B^\circ_2[g]$ sets.