A subset $A$ of a finite abelian group is called $(k,\ell)$-sum-free if $kA \cap \ell A=\emptyset.$ 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 $1 \leq k <\ell$ and a compact abelian group $G$, let $$\lambda_{ k,\ell}(G)=\sup\{ \mu(A): kA \cap \ell A =\emptyset \}$$ 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.
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