For any subset \(\mathcal A\) of \(\mathbb{N}\) let \(k{\mathcal A}\) denote the set of integers \(n = a_ 1 + \dots + a_ k\), with \(a_ i \in {\mathcal A}\), and let \({\mathcal A}(N) = {\mathcal A} \cap [1,N]\). It is conjectured that if the density \(\#{\mathcal A}(N)/N\) is not too small, then \({\mathcal A}(N)\) will contain a long arithmetic progression. Here it is shown that if \(\#{\mathcal A}(N)/N \geq (\log N)^{-\alpha}\), where \(0 < \alpha < {1\over 3}\), and \(N > N(\alpha)\), then \(3{\mathcal A}(N)\) contains an arithmetic progression of length \(\text{exp}(c(\log N)^{1-3\alpha})\). A corresponding result for \(2{\mathcal A}(N)\) has been given by J. Bourgain [A tribute to P. Erdős, 105-109 (1990; Zbl 0715.11006)].
The paper goes on to show that, if a certain ‘local’ condition (that \(\mathcal A\) covers certain congruence classes) is satisfied, then \(A\) is an asymptotic basis of order 4. The proofs are an elegant combination of combinatorial arguments with the use of exponential sums. The paper concludes with an example showing that the first result cannot be too far from the truth.