Kneser- and Jin-type inverse theorems in discrete abelian groups
John T. Griesmer
TL;DR
This work develops a Kneser- and Jin-type inverse theory for sumsets in general discrete abelian groups by leveraging finitely additive translation-invariant means, Følner-analytic densities, and upper Banach density. The authors establish a stabilizer framework (the KJ-stabilizer) that governs when a sumset inequality $ ilde{m}(A+B)< ilde{m}(A)+ ilde{m}(B)$ or $d^{*}(A+B)<d^{*}(A)+d^{*}(B)$ forces periodicity modulo a finite-index subgroup, with precise equalities like $ ilde{m}(A+B)= ilde{m}(A+K)+ ilde{m}(B+K)- ilde{m}(K)$. They extend Kneser-type results from $oldsymbol{\mathbb{Z}}$ to arbitrary groups along Følner sequences and nets, via a robust analytic framework built on invariant means, Choquet decomposition, Bohr compactifications, and a Hilbert-space $L^{2}(m)$ associated to a mean. The main contributions include a general Main One-Mean Strict theorem and a Folner-net extension (Main Folner Strict), which together generalize Bihani–Jin and Jin results to countable and uncountable discrete abelian groups, with finite-index stabilizers producing exact density equalities after passing to the quotient by $K$. The results significantly broaden sumset structure theorems beyond countable or purely countable settings, offering stable, finite-periodic descriptions of when subadditivity gaps occur, and establishing tools (Choquet decompositions, level-set analysis in $L^{2}(m)$, and Følner net extraction) that may influence future inverse problems in ergodic combinatorics and additive number theory.
Abstract
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 $Γ$.