Strong characterizing sequences of countable groups

TL;DR

This paper extends the characterization of countable subgroups of the circle group using sequences with specific convergence properties, introducing new methods and broader applicability.

Contribution

It generalizes previous results to all countable subgroups of the circle group and incorporates small powers of norms, using novel filter-based techniques.

Findings

Characterization of countable subgroups via sequences with summability properties

Extension of previous results to arbitrary countable subgroups of 2

Introduction of methods involving filter characterizations of subgroups

Abstract

Andr\'as Bir\'o and Vera S\'os prove that for any subgroup $G$ of $\T$ generated freely by finitely many generators there is a sequence $A\subset \N$ such that for all $\beta \in \T$ we have ($\|.\|$ denotes the distance to the nearest integer) $$\beta\in G \Rightarrow \sum_{n\in A} \| n \beta\| < \infty,\quad \quad \quad \beta\notin G \Rightarrow \limsup_{n\in A, n \to \infty} \|n \beta\| > 0. $$ We extend this result to arbitrary countable subgroups of $\T$. We also show that not only the sum of norms but the sum of arbitrary small powers of these norms can be kept small. Our proof combines ideas from the above article with new methods, involving a filter characterization of subgroups of $\T$.