×

On the maximal density of sum-free sets. (English) Zbl 0962.11013

For a set \(A\subseteq \mathbb N\) the subset sums of \(A\) are defined by \(P(A)=\{\sum_{a\in B} a : B\subseteq A\), \(1\leq |B|< \infty\}.\) Furthermore let \(Q(A)=\{\sum_{a\in B} a : B\subseteq A, 2\leq |B|< \infty\}.\) A set \(A\) is said to be sum-free if \(A\cap Q(A)=\emptyset.\) Let \(A(n)\) be the counting function of \(A\), i.e. \(A(n)=|A\cap \{1,2,\dots,n\}|.\)
In the present paper the authors prove the following results: If \(A\) is a set of natural numbers with \(A(n)>402\sqrt{n\cdot\log n}\) then \(P(A)\) contains an infinite arithmetic progression.
This result is an improvement of a well-known result of Folkman (and also proved by the reviewer). Furthermore the authors also improve a result of Deshouillers, Erdős and Melfi, proving if \(A\subseteq\mathbb N\) is a sum-free set then for large \(n\) \(A(n)\leq 403\sqrt{n\cdot\log n}.\) On the other hand it is proved there exists a sum-free set \(B\) such that \(B(n)\geq n^{1/2}/\log n^{1/2+\varepsilon}.\)

MSC:

11B75 Other combinatorial number theory
PDFBibTeX XMLCite
Full Text: DOI EuDML