On sums and products of distinct numbers
type: Item
instance of: scholarly article
title: On sums and products of distinct numbers
zbMATH Open document ID: 1060.11012
author: Mei-Chu Chang
published in: Journal of Combinatorial Theory. Series A
publication date: 2004-08-06T00:00:00Z
review text: 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.
reviewed by: Imre Z. Ruzsa
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 05D05
zbMATH DE Number: 2083276
zbMATH Keywords: sumset
zbMATH Keywords: product set
MaRDI profile type: Publication
full work available at URL: j.jcta.2003.10.004
OpenAlex ID: W2070170532
cites work: The Erdős-Szemerédi problem on sum set and product set
cites work: Factorization in generalized arithmetic progressions and application to the Erdős-Szemerédi sum-product problems
cites work: Q3041274
cites work: Q3215325
cites work: Q4895030
cites work: Q5579008
DOI: 10.1016/J.JCTA.2003.10.004
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description: scientific article; zbMATH DE number 2083276