Long Arithmetic Progressions in Small Sumsets

Let A, B ⊆ Z be finite, nonempty subsets such that maxB − minB ≤ maxA − minA, gcd(A+B−c) = 1, for some c ∈ A+B, and |A+B| ≤ |A|+2|B|−3−δ(A,B), where δ(A,B) is 1 if x + A ⊆ B for some x ∈ Z, and is 0 otherwise. Assume one of the following conditions holds true: • maxA−minA ≤ |A| + |B|− 3, • gcd(A− a) ≤ 2, for some a ∈ A, • |A + B| ≤ 2|A| + |B|− 3− δ(B,A). Then A+B contains a (|A|+ |B|−1)–term arithmetic progression with difference 1.