Infinite unrestricted sumsets in subsets of abelian groups with large density
Dimitrios Charamaras, Ioannis Kousek, Andreas Mountakis, Tristán Radić
TL;DR
This paper extends Erdős-type infinite sumset results from $\mathbb{N}$ to countable abelian groups $G$ with ${[G:2G]}<\infty$ and finite $\ker(D)$, establishing density thresholds with respect to q.i.d. Følner sequences: if $\overline{d}_{\Phi}(A) > 1 - \,\frac{\alpha_{\Phi}}{\ell + r}$ (or the related bounds for restricted configurations), then $A$ contains a shifted or unrestricted infinite sumset of the form ${t+B+B}$ or ${B+B}$ with $B$ infinite. The authors translate the combinatorial problem into a dynamical one via a Furstenberg-type correspondence and Erdős progressions, prove the dynamical analogue, and then transfer it back to the original setting. They prove optimality of the density bounds by constructing detailed counterexamples across several cases of $(\ell,r)$ using groups like $\mathbb{Z}^d$, dyadic tori, and their products, and they show the necessity of the underlying hypotheses (finite kernel, q.i.d., finite index). The paper also establishes that, for groups with ${[G:2G]}<\infty$ and finite $\ker(D)$, there exist Følner sequences that achieve the maximal possible quasi-invariant ratio $\alpha_G=\min\{1, r/\ell\}$. Overall, the work bridges combinatorial number theory and ergodic theory to characterize when high-density sets in abelian groups must contain large unrestricted sumsets and identifies when the bounds can be attained or are sharp.
Abstract
Let $(G,+)$ be a countable abelian group such that the subgroup $\{g+g\colon g\in G\}$ has finite index and the doubling map $g\mapsto g+g$ has finite kernel. We establish lower bounds on the upper density of a set $A\subset G$ with respect to an appropriate Følner sequence, so that $A$ contains a sumset of the form $\{t+b_1+b_2\colon b_1,b_2\in B\}$ or $\{b_1+b_2\colon b_1,b_2\in B\}$, for some infinite $B\subset G$ and some $t\in G$. Both assumptions on $G$ are necessary for our results to be true. We also characterize the Følner sequences for which this is possible. Finally, we show that our lower bounds are optimal in a strong sense.