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In this paper we show that every set $A \subset \mathbb {N}$ with positive density contains $B+C$ for some pair $B,C$ of infinite subsets of $\mathbb {N}$, settling a conjecture of Erd\H os. The proof features two different decompositions of an arbitrary bounded sequence into a structured component and a pseudo-random component. Our methods are quite general, allowing us to prove a version of this conjecture for countable amenable groups.
Joel Moreira 1 ; Florian K. Richter 1 ; Donald Robertson 2
@article{10_4007_annals_2019_189_2_4,
author = {Joel Moreira and Florian K. Richter and Donald Robertson},
title = {A proof of a sumset conjecture of {Erd\H{o}s}},
journal = {Annals of mathematics},
pages = {605--652},
publisher = {mathdoc},
volume = {189},
number = {2},
year = {2019},
doi = {10.4007/annals.2019.189.2.4},
mrnumber = {3919363},
zbl = {07041751},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.189.2.4/}
}
TY - JOUR AU - Joel Moreira AU - Florian K. Richter AU - Donald Robertson TI - A proof of a sumset conjecture of Erdős JO - Annals of mathematics PY - 2019 SP - 605 EP - 652 VL - 189 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.189.2.4/ DO - 10.4007/annals.2019.189.2.4 LA - en ID - 10_4007_annals_2019_189_2_4 ER -
%0 Journal Article %A Joel Moreira %A Florian K. Richter %A Donald Robertson %T A proof of a sumset conjecture of Erdős %J Annals of mathematics %D 2019 %P 605-652 %V 189 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.189.2.4/ %R 10.4007/annals.2019.189.2.4 %G en %F 10_4007_annals_2019_189_2_4
Joel Moreira; Florian K. Richter; Donald Robertson. A proof of a sumset conjecture of Erdős. Annals of mathematics, Tome 189 (2019) no. 2, pp. 605-652. doi: 10.4007/annals.2019.189.2.4
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