Long Arithmetic Progressions in Sets with Small Sumset

Abstract

Let $A,B\subseteq \mathbb{Z}$ be finite, nonempty subsets with $\min A=\min B=0$, and let $$δ(A,B)={\begin{array}{ll} 1 & \hbox{if} A\subseteq B, 0 & \hbox{otherwise.} If $\max B\le \max A\le |A|+|B|-3$ and \label{one}|A+B|\leq |A|+2|B|-3-δ(A,B), then we show $A+B$ contains an arithmetic progression with difference 1 and length $|A|+|B|-1$. As a corollary, if \eqref{one} holds, $\max (B)\le \max (A)$ and either $\gcd (A)=1$ or else $\gcd (A+B)=1$ and $|A+B|\le 2|A|+|B|-3$, then $A+B$ contains an arithmetic progression with difference 1 and length $|A|+|B|-1$.