Long Arithmetic Progressions in Sets with Small Sumset

Let A, B ⊆ Z be finite, nonempty subsets with minA = minB = 0, and let δ(A,B) = n 1 if A ⊆ B, 0 otherwise. If maxB ≤ maxA ≤ |A|+ |B| − 3 and (1) |A+B| ≤ |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 (1) holds, max(B) ≤ max(A) and either gcd(A) = 1 or else gcd(A+ B) = 1 and |A+B| ≤ 2|A|+ |B| − 3, then A+B contains an arithmetic progression with difference 1 and length |A|+ |B| − 1.