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Arithmetic progressions in sumsets. (English) Zbl 0728.11009

Let A,B\(\subset [1,N]\) be sets of integers, \(| A| =| B| =cN\). J. Bourgain proved [A tribute to Paul Erdös, 105-109 (1990; Zbl 0715.11006)] that \(A+B\) always contains an arithmetic progression of length exp(log N)\({}^{1/3-\epsilon}\). Here an upper bound is found. It is proved that for every \(\epsilon >0\) and every prime \(p>p_ 0(\epsilon)\) there is a symmetric set A of residues mod p such that \(| A| >(1/2-\epsilon)p\) and \(A+A\) contains no arithmetical progression of length exp(log p)\({}^{2/3+\epsilon}\). The 1/2 in the theorem is optimal: if \(| A| >p/2\), then \(A+A\) contains every residue.
The example is a set of the form \(A=\{x: f(x)>Q\}\), where \(f(x)=\sum^{k}_{j=1}\cos (2\pi a_ jx/p)\) is a trigonometric polynomial, and the numbers \(a_ j\) are selected randomly.
Reviewer: I.Z.Ruzsa

MSC:

11B25 Arithmetic progressions
11P55 Applications of the Hardy-Littlewood method
11L03 Trigonometric and exponential sums (general theory)

Citations:

Zbl 0715.11006
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