On sums and products of distinct numbers
For a set of \(k\) complex numbers we form the \(2^k\) formal subset sums and the \(2^k\) formal subset products. Let \(g(k) \) denote the minimum of the cardinality of the union of these two sets taken over all \(k\)-element sets. The main result of the paper asserts that \(g(k)\gg k^c\) for every constant \(c\). This was conjectured for sets of integers by \textit{P. Erdős} and \textit{E. Szemerédi} [Studies in Pure Mathematics, Mem. of P. Turan, 213--218 (1983; Zbl 0526.10011)], and proved by the author [Ann. Math. (2) 157, No. 3, 939--957 (2003; Zbl 1055.11017)]. The rate of growth of the exponent is not estimated; the reviewer thinks it would be very slow, in contrast to the author's result for integers where it grows as \( ( \log k)/ ( \log \log k)\), which is best possible. The argument used here is completely different. It is based on Freiman's theory of structure of sets with small sumsets and the author's estimate of equal products formed from a generalized arithmetic progression [Geom. Funct. Anal. 13, No. 4, 720--736 (2003; Zbl 1029.11006)]. The proof for the case of integers ``relies heavily on factorization into primes \dots and does not extend beyond the integer case'', writes the author. The reviewer disagrees, and thinks that such an extension is probably feasible.
scientific article; zbMATH DE number 2083276