Strong characterizing sequences for subgroups of compact groups (Q863294)

For a finitely generated subgroup \(\Gamma\) of a compact metrizable Abelian group \(G\) it is shown that there is a subset \(A\) of the dual group \(G^*\) of \(G\) such that \(\gamma\in\Gamma\) iff \(\sum_{a\in A}\nu(\| a\gamma\|)<\infty\). Here \(\nu\) is a weight function and \(\| x\| \) denotes the distance of \(x\) to the nearest integer. This was proved in [\textit{A. Biró} and \textit{V. T. Sós}, J. Number Theory 99, No. 2, 405--414 (2003; Zbl 1058.11047)] for the torus group. Conditions for different weight functions \(\nu\) are given, which guarantee that there is \(A\subset G^*\) which yield characterizing sequences of the above type for both weight functions.