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Award ID contains: 2054643

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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. ; ; ; (, 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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  3. ; ; ; (, 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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  4. ; ; ; (, Israel Journal of Mathematics)
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  5. ; (, Proceedings of the American Mathematical Society, Series B)
    Boshernitzan gave a decay condition on the measure of cylinder sets that implies unique ergodicity for minimal subshifts. Interest in the properties of subshifts satisfying this condition has grown recently, due to a connection with discrete Schrödinger operators, and of particular interest is how restrictive the Boshernitzan condition is. While it implies zero topological entropy, our main theorem shows how to construct minimal subshifts satisfying the condition, and whose factor complexity grows faster than any pre-assigned subexponential rate. As an application, via a theorem of Damanik and Lenz, we show that there is no subexponentially growing sequence for which the spectra of all discrete Schrödinger operators associated with subshifts whose complexity grows faster than the given sequence have only finitely many gaps. 
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  6. ; (, Monatshefte für Mathematik)
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  7. ; ; (, International Mathematics Research Notices)
    Abstract For a mixing shift of finite type, the associated automorphism group has a rich algebraic structure, and yet we have few criteria to distinguish when two such groups are isomorphic. We introduce a stabilization of the automorphism group, study its algebraic properties, and use them to distinguish many of the stabilized automorphism groups. We also show that for a full shift, the subgroup of the stabilized automorphism group generated by elements of finite order is simple and that the stabilized automorphism group is an extension of a free abelian group of finite rank by this simple group. 
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