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Creators/Authors contains: "Moreira, Joel"

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  1. ; ; ; (, Communications of the American Mathematical Society)
    We show that every set A A of natural numbers with positive upper Banach density can be shifted to contain the restricted sumset { b 1 + b 2 : b 1 , b 2 ∈<#comment/> B  and  b 1 ≠<#comment/> b 2 } \{b_1 + b_2 : b_1, b_2\in B \text { and } b_1 \ne b_2 \} for some infinite set B ⊂<#comment/> A B \subset A . 
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  2. ; ; ; (, Journal of the American Mathematical Society)
    Motivated by questions asked by Erdős, we prove that any set A ⊂<#comment/> N A\subset \mathbb {N} with positive upper density contains, for any k ∈<#comment/> N k\in \mathbb {N} , a sumset B 1 + ⋯<#comment/> + B k B_1+\cdots +B_k , where B 1 B_1 , …, B k ⊂<#comment/> N 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 k=2 . 
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    Full Text Available 
  3. ; ; (, Annals of Mathematics)
    Full Text Available 
  4. ; ; (, Annals of mathematics)
    In this paper we show that every set A⊂ℕ with positive density contains B+C for some pair B,C of infinite subsets of 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. 
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    Accepted Manuscript Full Text Available