Infinite sumsets in sets with positive density

Kra, Bryna; Moreira, Joel; Richter, Florian; Robertson, Donald

doi:10.1090/jams/1030

Infinite sumsets in sets with positive density
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by Bryna Kra, Joel Moreira, Florian K. Richter and Donald Robertson;

J. Amer. Math. Soc. 37 (2024), 637-682

DOI: https://doi.org/10.1090/jams/1030

Published electronically: August 11, 2023

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Abstract:

Motivated by questions asked by Erdős, 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 $B_1$, …, $B_k\subset \mathbb {N}$ 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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