Di Nasso, Mauro; Goldbring, Isaac; Jin, Renling; Leth, Steven; Lupini, Martino; Mahlburg, Karl On a sumset conjecture of Erdős. (English) Zbl 1365.11008 Can. J. Math. 67, No. 4, 795-809 (2015). Summary: Erdős conjectured that for any set \(A\subseteq \mathbb{N}\) with positive lower asymptotic density, there are infinite sets \(B,C \subseteq \mathbb{N}\) such that \(B+C \subseteq A\). We verify Erdős’ conjecture in the case that \(A\) has Banach density exceeding \(\frac{1}{2}\). As a consequence, we prove that, for \(A\subseteq \mathbb{N}\) with positive Banach density (a much weaker assumption than positive lower density), we can find infinite \(B,C \subseteq \mathbb{N}\) such that \(B+C\) is contained in the union of \(A\) and a translate of \(A\). Both of the aforementioned results are generalized to arbitrary countable amenable groups. We also provide a positive solution to Erdős’ conjecture for subsets of the natural numbers that are pseudorandom. Cited in 6 Documents MSC: 11B05 Density, gaps, topology 11B13 Additive bases, including sumsets 11P70 Inverse problems of additive number theory, including sumsets 28D15 General groups of measure-preserving transformations 37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010) Keywords:sumsets of integers; asymptotic density; amenable groups; nonstandard analysis PDFBibTeX XMLCite \textit{M. Di Nasso} et al., Can. J. Math. 67, No. 4, 795--809 (2015; Zbl 1365.11008) Full Text: DOI arXiv