On quasi-convex null sequences in infinite cyclic groups (Q723290)

A subset \(A\) of a topological abelian group \((G,\mathcal T)\) is called quasi-convex if for every \(x\in G\setminus A\) there exists a continuous character \(\chi\) such that \(\chi(A)\subseteq \mathbb T_+\) and \(\chi(x)\notin \mathbb T_+,\) where \(\mathbb T_+=([\frac{-1}{4},\frac{1}{4}]+\mathbb Z)/\mathbb Z\subseteq\mathbb R/\mathbb Z\). A sequence \((x_n)_{n\in\mathbb N}\) in a topological abelian group is called quasi-convex null sequence if the set \(\{0\}\cup \{\pm x_n:n\in\mathbb N\}\) is quasi-convex and \(\lim_nx_n=0.\) In the main Theorem there are constructed some quasi-convex subsets of the torus \(\mathbb R/\mathbb Z\). Using these subsets it is proved that there are \(\mathfrak{c}\) many irrational numbers \(\beta\in]-\frac{1}{2},\frac{1}{2}[\) such that the cyclic subgroup \(\langle \beta+\mathbb Z\rangle\) of \(\mathbb R/\mathbb Z\) generated by \(\beta+\mathbb Z\) contains a non-trivial quasi-convex null sequence.