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Award ID contains: 2348315

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  1. ; ; ; (, Bulletin of the American Mathematical Society)
    In contrast to finite arithmetic configurations, relatively little is known about which infinite patterns can be found in every set of natural numbers with positive density. Building on recent advances showing infinite sumsets can be found, we explore numerous open problems and obstructions to finding other infinite configurations in every set of natural numbers with positive density. 
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    Free, publicly-accessible full text available July 1, 2026
  2. ; ; (, Mathematische Zeitschrift)
    Free, publicly-accessible full text available January 1, 2026
  3. ; ; (, Proceedings of the American Mathematical Society)
    A subset R R of integers is a set of Bohr recurrence if every rotation on T d \mathbb {T}^d returns arbitrarily close to zero under some non-zero multiple of R R . We show that the set { k ! 2 m 3 n :<#comment/> k , m , n ∈<#comment/> N } \{k!\, 2^m3^n\colon k,m,n\in \mathbb {N}\} is a set of Bohr recurrence. This is a particular case of a more general statement about images of such sets under any integer polynomial with zero constant term. We also show that if P P is a real polynomial with at least one non-constant irrational coefficient, then the set { P ( 2 m 3 n ) :<#comment/> m , n ∈<#comment/> N } \{P(2^m3^n)\colon m,n\in \mathbb {N}\} is dense in T \mathbb {T} , thus providing a joint generalization of two well-known results, one of Furstenberg and one of Weyl. 
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    Free, publicly-accessible full text available January 1, 2026
  4. ; ; ; (, 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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