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arXiv · 1307.0767

On a sumset conjecture of Erdős

Mauro Di Nasso·Isaac Goldbring·Renling Jin·Steven Leth·Martino Lupini·Karl Mahlburg
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Abstract

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.

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Mauro Di Nasso, Isaac Goldbring, Renling Jin, Steven Leth, Martino Lupini, Karl Mahlburg. 2014-04-23. On a sumset conjecture of Erdős. https://doi.org/10.4153/cjm-2014-016-0

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