Bohr sets in sumsets I: Compact abelian groups

Abstract

Let $G$ be a compact abelian group and $\ddot{I}{•}_1,\ddot{I}{•}_2,\ddot{I}{•}_3$ be continuous endomorphisms on $G$. Under certain natural assumptions on the $\ddot{I}{•}_i$'s, we prove the existence of Bohr sets in the sumset $\ddot{I}{•}_1(A)+\ddot{I}{•}_2(A)+\ddot{I}{•}_3(A)$, where $A$ is either a set of positive Haar measure, or comes from a finite partition of $G$. The first result generalizes theorems of Bogolyubov and Bergelson-Ruzsa. As a variant of the second result, we show that for any partition $\mathbb{Z}={\bigcup }_{i=1}^r{A}_i$, there exists an $i$ such that $A_i-A_i+s{A}_i$ contains a Bohr set for any $s\in \mathbb{Z}\setminus \{0\}$. The latter is a step toward an open question of Katznelson and Ruzsa.

37 pages. Discrete Analysis 2025:11