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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.19014 |
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Table of Contents:
- We characterize the pairs of sets $A, B$ in an arbitrary (countable or uncountable) discrete abelian group $Γ$ satisfying $\tilde{m}(A+B)<\tilde{m}(A)+\tilde{m}(B)$, where $\tilde{m}$ is an arbitrary finitely additive translation-invariant probability measure on $Γ$, extending M.~Kneser's theorem on Haar measure in compact abelian groups. We then characterize, for an arbitrary Følner sequence or Følner net $\mathbf F=(F_{i})_{i\in I}$ on $Γ$, those $A$, $B$ satisfying $\underline{d}_{\mathbf F}(A+B)<\underline{d}_{\mathbf F}(A)+\underline{d}_{\mathbf F}(B)$, where $\underline{d}_{\mathbf F}(C):=\liminf_{i\in I} |C\cap F_{i}|/|F_{i}|$. This extends Kneser's theorem on lower asymptotic density in $\mathbb N$. We also generalize theorems of Prerna Bihani and Renling Jin by characterizing pairs $A$, $B$ satisfying $d^{*}(A+B)<d^{*}(A)+d^{*}(B)$, where $d^{*}$ is upper Banach density on $Γ$.