On the density of sets containing no three distinct numbers with all their sums
type: Item
instance of: scholarly article
title: On the density of sets containing no three distinct numbers with all their sums
zbMATH Open document ID: 0967.11010
DOI: 10.1007/BF03322437
author: Tomasz Schoen
published in: Results in Mathematics
publication date: 2000-08-31T00:00:00Z
review text: Let \(A\) be a subset of positive integers and let \(X\subseteq A.\) Denote by \(P(X)\) the subset sums of \(X,\) i.e. let \(P(X)=\{\sum_{y\in Y}y : Y\subseteq X, Y\neq \emptyset\}.\) Choi, Erdős and Szemerédi proved that for every \(k\in \mathbb N,\) there exists \(\alpha_k,\) \(0<\alpha_k<1,\) such that if \(\bar d(A)>1-\alpha_k\) then there exists an \(X\subseteq A;\) \(|X|=k\) and \(P(X)\subseteq A.\) It is well-known (an easy exercise) that \(\alpha_2=1/2.\) In the present paper the author investigates the first nontrivial case proving \(\alpha_3=1/3.\) The proof is combinatorial and uses a result of Luczak and Schoen on sum-free set \(A'\) with \(\bar d(A')>1/3.\)
Mathematics Subject Classification ID: 11B75
zbMATH DE Number: 1498584
zbMATH Keywords: density
zbMATH Keywords: sum-free sets
reviewed by: Norbert Hegyvári
MaRDI profile type: Publication
cites work: Q4879099
cites work: A note on the density of sum-free sets
cites work: On infinite sum-free sets of natural numbers
cites work: On strongly sum-free subsets of abelian groups
description: scientific article; zbMATH DE number 1498584