Finite addition theorems. II
type: Item
instance of: scholarly article
title: Finite addition theorems. II
zbMATH Open document ID: 0808.11011
published in: Journal of Number Theory
publication date: 1994-09-28T00:00:00Z
review text: [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)].
Mathematics Subject Classification ID: 11B13
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 11B25
Mathematics Subject Classification ID: 11P99
Mathematics Subject Classification ID: 11N25
zbMATH DE Number: 645293
zbMATH Keywords: sumsets
zbMATH Keywords: sums with distinct summands
zbMATH Keywords: arithmetic progression
zbMATH Keywords: long arithmetic progression
zbMATH Keywords: subset sums
author: András Sárközy
reviewed by: Imre Z. Ruzsa
MaRDI profile type: Publication
full work available at URL: jnth.1994.1062
OpenAlex ID: W2076729819
DOI: 10.1006/JNTH.1994.1062
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description: scientific article; zbMATH DE number 645293