Bohr sets in sumsets in countable abelian groups

Abstract: A \textit{Bohr set} in an abelian topological group $G$ is a subset of the form \[ B(K, \epsilon) = \{ g \in G: |\chi(g) - 1| < \epsilon \, \forall \chi \in K \} \] where $K$ is a finite subset of the dual group $\widehat{G}$. A classical theorem of Bogolyubov says that if $A \subset \mathbf{Z}$ has positive upper density $\delta$, then $A+A-A-A$ contains a Bohr set $B(K, \epsilon)$ where $|K|$ and $\epsilon$ depend only on $\delta$. While the same statement for $A-A$ is not true (a result of K\v{r}\'i\v{z}), Bergelson and Ruzsa proved that if $r+s+t=0$, then $rA + sA+tA$ contains a Bohr set (here $rA = \{ ra: a \in A \}$). We investigate this phenomenon in more general groups $G$, where $rA, sA, tA$ are replaced by images of $A$ under certain endomomorphisms of $G$. It is also natural to ask for partition analogues of the Bergelson-Ruzsa theorem. In CANT 2021, I discussed our results in compact abelian groups (generalizations of $\mathbf{R} /\mathbf{Z}$). \ In this talk, I will discuss our progress on countable discrete abelian groups (generalizations of $\mathbf{Z}$). The key ingredients are certain transference principles which allow us to transfer the results from compact groups to discrete countable groups. This talk is based on joint works with Anh Le, and with Anh Le and John Griesmer.

number theory

Audience: researchers in the discipline

Combinatorial and additive number theory (CANT 2022)