Finite addition theorems. II
[For part I, cf. ibid. 32, 114-130 (1989; Zbl 0674.10042).] This paper is devoted to the study of sets of sums with distinct summands. The main result asserts that if \(A\subset [1,N]\) is a set of integers, \(| A|\gg (N\log N)^{1/2}\), then (for every ``reasonably sized'' \(M\)) we can find \(d\), \(s\) such that \(d\ll N/| A|\), \(s\ll M/| A|\) and there is an arithmetic progression of length \(M\) and difference \(d\) whose terms can be written as sums of \(s\) distinct elements of \(A\). This result is nearly best possible (the logarithm in the bound for \(| A|\) may be superfluous, and the values of the constants and the bounds for \(M\) which are not quoted here may not be optimal). Various applications of this theorem are given. A long arithmetic progression is found in the set of all subset sums (this result was also obtained, by different methods, by G. Freiman). This is used to give estimates for the size of \(A\) if there is no \(r\)-th power among the subset sums, which improves previous results of \textit{E. Lipkin} [Acta Arith. 52, 353-366 (1989; Zbl 0691.10042)] and \textit{N. Alon} and \textit{G. Freiman} [Combinatorica 8, 297-306 (1988; Zbl 0666.10035)]. The analogous problem for integers of the form prime \(-1\) is also studied; here the result is much weaker than expected, because this is closely related to the problem of the smallest prime in an arithmetic progression. [Part III, cf. Publ. Math. Orsay 92-01, 105-122 (1992; Zbl 0780.11008)].
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