Characterized subgroups related to some non-arithmetic sequence of integers (Q6591076)

Let \(\mathbb T=\mathbb R/\mathbb Z\) denote the circle group in additive notation, let \(\pi\colon\mathbb R\to \mathbb T\) be the canonical projection and denote \(\bar x=\pi(x)\) for every \(x\in\mathbb R\). For a sequence \((u_n)\) of integers, \(t_{(u_n)}(\mathbb T)=\{\bar x\in \mathbb T:u_n\bar x\to 0\}\).\N\NA sequence \((a_n)\) of integers is \textit{arithmetic} if it is strictly increasing, \(a_0=1\) and \(a_n\mid a_{n+1}\) for every \(n\in\mathbb N\). For every \(n\in\mathbb N_+\), let \(b_n=\frac{a_n}{a_{n+1}}\). Then \((a_n)\) is called \textit{\(b\)-bounded} if \((b_n)\) is bounded.\N\NGiven an arithmetic sequence \((a_n)\), for every \(x\in[0,1)\) there exists a unique sequence \((c_n)\) in \(\mathbb N\) such that \(x=\sum_{n\in\mathbb N_+}\frac{c_n}{a_n}\), with \(c_n<b_n\) for every \(n\in\mathbb N_+\) and \(c_n<b_n-1\) for infinitely many \(n\in\mathbb N_+\). The \textit{support} of \(x\in[0,1)\) with respect to \((a_n)\) is \(\mathrm{supp}_{(a_n)}(x)=\{n\in\mathbb N_+: c_n\neq0\}\).\N\N\NInspired by similar constructions by \textit{A. Bíró} et al. [Stud. Sci. Math. Hung. 38, 97--113 (2001; Zbl 1006.11038)] and \textit{D. Dikranjan} and \textit{K. Kunen} [J. Pure Appl. Algebra 208, No. 1, 285--291 (2007; Zbl 1109.22002)], fixed an arithmetic sequence \((a_n)\), the authors introduce the strictly increasing sequence \((d_n)\) defined by the set \(\bigcup_{k\in\mathbb N}\{ra_k:1\leq r\leq b_{k+1}\}\). Since \((a_n)\) is a subsequence of \((d_n)\), clearly \(t_{(d_n)}(\mathbb T)\subseteq t_{(a_n)}(\mathbb T)\).\N\NOne of the main result of the paper is that given an arithmetic sequence \((a_n)\), \(t_{(d_n)}(\mathbb T)=t_{(a_n)}(\mathbb T)\cap \mathbb Q/\mathbb Z\), that is, \(t_{(d_n)}(\mathbb T)\) is the torsion part of \(t_{(a_n)}(\mathbb T)\).\N\NAnother fundamental theorem states that, given an arithmetic sequence \((a_n)\) and \(x\in[0,1)\), then \(\bar x\in t_{(d_n)}(\mathbb T)\) if and only if \(\mathrm{supp}_{(a_n)}(x)\) is finite. A consequence of this result is that \(t_{(d_n)}(\mathbb T)\) is a countable unbounded torsion subgroup of \(\mathbb T\), so in particular it is an \(F_{\sigma}\) subgroup. Moreover, \(t_{(a_n)}(\mathbb T)=t_{(d_n)}(\mathbb T)\) precisely when \((a_n)\) is \(b\)-bounded and it is known that in this case \(t_{(a_n)}(\mathbb T)\) is countable and torsion.