Convexity and sumsets
type: Item
instance of: scholarly article
title: Convexity and sumsets
zbMATH Open document ID: 0998.11010
author: György Elekes
author: Melvyn B. Nathanson
author: Imre Z. Ruzsa
published in: Journal of Number Theory
publication date: 2001-01-11T00:00:00Z
review text: Let \(A\) be an \(n\) element subset of real numbers, and let \(f\) be a strictly convex or concave function defined on an interval containing \(A\). Let \(S=\{(a,f(a)): a\in A\}\subseteq \mathbb R^2\). The authors prove that for every finite set \(T\) \[ |S+T|\geq c \min\{n|T|, n^{3/2}|T|^{1/2}\}, \] where \(c\) is an absolute constant, \(0<c<1\). A generalization is also proved for a repeated addition of \(k\) identical sets \(S*k:=S+S+\dots +S\) (\(k\) times). They prove: for every \(k\geq 1\) \[ |S*k+T|\geq c'n^{3(1-2^{-k}})|T|^{2^{-k}} \quad\text{if }|T|\geq n, \] and \[ |S*k+T|\geq c'n^{3-2^{-(k-2)}}|T|^{2^{-k-1}}\quad\text{if }|T|<n, \] where \(c'\) is an absolute constant \(0<c'<1\). As a corollary they improve an earlier result of \textit{P. Erdős} and \textit{E. Szemerédi} [Studies in pure mathematics, Mem. of P. Turan, 213--218 (1983; Zbl 0526.10011)] and a result of the reviewer.
reviewed by: Norbert Hegyvári
Mathematics Subject Classification ID: 11B05
zbMATH DE Number: 1515454
zbMATH Keywords: sumsets
zbMATH Keywords: addition of sets
MaRDI profile type: Publication
full work available at URL: jnth.1999.2386
OpenAlex ID: W1998666916
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DOI: 10.1006/JNTH.1999.2386
description: scientific article; zbMATH DE number 1515454