On a Sumset Conjecture of Erdős

Nasso, Mauro Di ; Goldbring, Isaac ; Jin, Renling ; Leth, Steven ; Lupini, Martino ; Mahlburg, Karl

Canadian journal of mathematics, Tome 67 (2015) no. 4, pp. 795-809

Résumé

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 where $A$ 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.

Détail

DOI : 10.4153/CJM-2014-016-0

Mots-clés : 11B05, 11B13, 11P70, 28D15, 37A45, sumsets of integers, asymptotic density, amenable groups, nonstandard analysis

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