On sum-free subsets of the torus group

Abstract

Establishing the structure of dense sum-free subsets of the torus group $\mathbb{R}/\mathbb{Z}$, we find an absolute constant $\alpha_0<1/3$ such that for any sum-free subset $A\subseteq\mathbb{R}/\mathbb{Z}$ with the inner measure $\mu(A)>\alpha_0$ there exists an integer $q\ge 1$ so that $$A \subseteq \bigcup_{j=0}^{q-1}[ \frac{j+\mu(A)}q, \frac{j+1-\mu(A)}q ].$$