Arithmetic progressions in sumsets

1. Introduction. Let A,B ⊂ [1,N] be sets of integers, |A|=|B|=cN. Bourgain [2] proved that A+B always contains an arithmetic progression of length ${\displaystyle {\exp \left(\log N\right)}^{1/3-\epsilon }}$. Our aim is to show that this is not very far from the best possible. Theorem 1. Let ε be a positive number. For every prime p > p₀(ε) there is a symmetric set A of residues mod p such that |A| > (1//2-ε)p and A + A contains no arithmetic progression of length (1.1)} ${\displaystyle {\exp \left(\log p\right)}^{2/3+\epsilon }}$. A set of residues can be used to get a set of integers in an obvious way. Observe that the 1/2 in the theorem is optimal: if |A|>p//2, then A+A contains every residue. Acknowledgement. I profited much from discussions with E. Szemerédi; he directed my attention to this problem and to Bourgain's paper.