Sharp density conditions for infinite $B+B$ sumsets in abelian groups

Motivated by recent results \cite{charamaras_kousek_mountakis_radic2025BBingroups} on infinite sumsets of the form $B+B=\{b_1+b_2:b_1,b_2\in B\}$ in large subsets of abelian groups, and an old problem of Owings \cite[Problem E2494]{Owing_problems} about the partition regularity of $B+B$ in $2$ colours, we show the following theorem. Let $(G,+)$ be a countable abelian group such that the subgroup $\{g+g\colon g\in G\}$ has finite index and the doubling map $D: g\mapsto g+g$ has finite kernel. Let also $\Phi=(\Phi_N)_{N}$ be any Folner sequence in $G$ and $\Phi/2=(D^{-1}(\Phi_N))_{N}$. Then, if $A\subset G$ is such that $d_{\Phi}(A)+d_{\Phi/2}(A)>1$, there is an infinite set $B\subset G$ and some $t\in G$ for which $t+B+B\subset A$.
We prove that this result implies the main theorem in \cite{charamaras_kousek_mountakis_radic2025BBingroups}, and construct an example to show the reverse implication does not hold. Moreover, we show that our main theorem is optimal in a strong sense. Namely, for any countable abelian group $(G,+)$ with the aforementioned assumptions -- which are necessary -- there exists a Folner sequence $\Phi$ and a set $A\subset G$ so that $d_{\Phi}(A)+d_{\Phi/2}(A)=1$, but there is no infinite set $B\subset G$ and $t\in G$ for which $t+B+B\subset A$.
Finally, we relate the optimality of our main result in the integer setting to Owings' problem and present some other considerations around this.