Bohr sets in sumsets I: Compact abelian groups

Abstract

Let $G$ be a compact abelian group and $ϕ_1, ϕ_2, ϕ_3$ be continuous endomorphisms on $G$. Under certain natural assumptions on the $ϕ_i$'s, we prove the existence of Bohr sets in the sumset $ϕ_1(A) + ϕ_2(A) + ϕ_3(A)$, 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 $A_i - A_i + sA_i$ 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.