The largest $(k,\ell )$-sum-free sets in compact abelian groups

Abstract

A subset $A$ of a finite abelian group is called $(k,\ell )$-sum-free if $kA\cap \ell A=\varnothing .$ In this paper, we extend this concept to compact abelian groups and study the question of how large a measurable $(k,\ell )$-sum-free set can be. For integers $1\le k<\ell$ and a compact abelian group $G$, let $${\lambda }_{k,\ell }(G)=\sup \{\mu (A):kA\cap \ell A=\varnothing \}$$ be the maximum possible size of a $(k,\ell )$-sum-free subset of $G$. We prove that if $G=\mathbb{I}\times M$, where $\mathbb{I}$ is the identity component of $G$, then $${\lambda }_{k,\ell }(G)=\max \left\{{\lambda }_{k,\ell }(M),{\lambda }_{k,\ell }(\mathbb{I})\right\}.$$ Moreover, if $\mathbb{I}$ is nontrivial, then ${\lambda }_{k,\ell }(\mathbb{I})=\frac{1}{k+\ell }$. Finally, we discuss how this problem motivates a new framework for studying $(k,\ell )$-sum-free sets in finite groups.