Blocks of consecutive integers in sumsets (A + B)t

Abstract

Let A, B ⊆ {1, …, n}. For m ∈ Z, let rA,B(m) be the cardinality of the set of ordered pairs (a, b) ∈ A × B such that a + b = m. For t ≥ 1, denote by (A + B)t the set of the elements m for which rA, B(m) ≥ t. In this paper we prove that for any subsets A, B ⊆ {1, …, n} such that |A| + |B| ≥ (4n + 4t − 3)/3, the sumset (A + B)t contains a block of consecutive integers with the length at least |A| + |B| − 2t + 1, and that (a) for any two subsets A and B of {1, …, n} such that |A| + |B| ≥ (4n)/3, there exists an arithmetic progression of length n in A + B; (b) for any 2 ≤ r ≤ (4n − 1)/3, there exist two subsets A and B of {1, …, n} with |A| + |B| = r such that any arithmetic progression in A + B has the length at most (2n − 1)/3 + 1.