Long Arithmetic Progressions in Sets with Small Sumset

TL;DR

This paper proves that under certain small sumset conditions, the sum of two finite integer sets contains a long arithmetic progression with difference 1, extending previous additive combinatorics results.

Contribution

It establishes new conditions ensuring the sumset contains a long arithmetic progression, generalizing prior theorems in additive number theory.

Findings

Sumsets with small size relative to original sets contain long arithmetic progressions.

Provides conditions involving gcd and sumset size for the existence of such progressions.

Extends classical results in additive combinatorics to broader set configurations.

Abstract

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