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A proof of a sumset conjecture of Erdős. (English) Zbl 1407.05236
Summary: 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 Erdős. 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.

MSC:
05D10 Ramsey theory
11P70 Inverse problems of additive number theory, including sumsets
37A99 Ergodic theory
46C99 Inner product spaces and their generalizations, Hilbert spaces
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