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 . 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). 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.
@article{bwmeta1.element.bwnjournal-article-aav60i2p191bwm,
author = {Imre Z. Ruzsa},
title = {Arithmetic progressions in sumsets},
journal = {Acta Arithmetica},
volume = {58},
year = {1991},
pages = {191-202},
zbl = {0728.11009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav60i2p191bwm}
}
Imre Z. Ruzsa. Arithmetic progressions in sumsets. Acta Arithmetica, Tome 58 (1991) pp. 191-202. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav60i2p191bwm/
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