Infinite Sumsets in Sets with Positive Density

Abstract

Motivated by questions asked by Erdos, we prove that any set $A\subset \mathbb{N}$ with positive upper density contains, for any $k\in \mathbb{N}$, a sumset $B_1+\cdots +B_k$, where $B_1,\dots ,B_k\subset \mathbb{N}$ are infinite. Our proof uses ergodic theory and relies on structural results for measure preserving systems. Our techniques are new, even for the previously known case of $k=2$.

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