Long Arithmetic Progressions in Sets with Small Sumset

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dc.creatorBardaji, Itziar
dc.creatorGrynkiewicz, David J.
dc.date2009-04-22
dc.date.accessioned2026-07-07T13:07:20Z
dc.date.available2026-07-07T13:07:20Z
dc.descriptionLet $A, B\subseteq \mathbb{Z}$ 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 $\max B\leq \max A\leq |A|+|B|-3$ 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, $\max(B)\leq \max(A)$ 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 $|A|+|B|-1$.
dc.identifierhttps://arxiv.org/abs/0904.3514
dc.identifierhttp://arxiv.org/abs/0904.3514
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/228055
dc.subjectNumber Theory
dc.subjectCombinatorics
dc.subject11P70, 11B25
dc.titleLong Arithmetic Progressions in Sets with Small Sumset
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