Characterizations of groups generated by Kronecker sets
Let \(\mathbb R\) denote the additive group of real numbers, \(\mathbb Z\) its subgroup consisting of the integers and \(T\) the torus \(\mathbb R/\mathbb Z\). A set \(K\subseteq T\) is called a ``Kronecker set'' if it is nonempty, compact, and for every continuous function \(f:K\rightarrow T\) and \(\delta>0\) there is an \(n\in{\mathbb Z}\) such that \[ \max_{\alpha\in K}\|f(\alpha)-n\alpha\|<\delta. \] (If \(x\in{\mathbb R},\) then \(\|x\|\) denotes the distance of \(x\) to the nearest integer.) Let \(K\) be a Kronecker set and \(G\) the subgroup of \(T\) generated by \(K.\) The author proves the following characterization of \(G\) by subsets \(A\) of \({\mathbb Z}^2.\) Put \(\ell(0)=0,\) \(\ell(x)=-(\log_2x)^{-1}\) if \(0<x<1/2,\) \(\ell(x)=1\) if \(x\geq1/2.\) Theorem. Let \(\varepsilon>0\) be an arbitrary positive real number. Then there is an infinite subset \(A\subseteq{\mathbb Z}^2\) (\(A\) depends on \(\varepsilon)\) such that for every \(\alpha\in G\) we have \[ \sum_{{\mathbf n}=(n_1,n_2)\in A}\ell^{1+\varepsilon}(\min(\|n_1\alpha\|,\|n_2\alpha\|))<\infty, \] and if \(\beta\in T\) satisfies \[ \min(\|n_1\beta\|,\|n_2\beta\|)<{1\over{10}} \] for all but finitely many \({\mathbf n}=(n_1,n_2)\in A,\) then \(\beta\in G.\) The author proves also that if \(K\) is uncountable, then this characterization is impossible by subsets \(A\) of \(\mathbb Z\).
scientific article; zbMATH DE number 5302615
scientific article