Statistically characterized subgroups of the circle (II): continued fractions

TL;DR

This paper explores statistically characterized subgroups of the circle group related to continued fractions, showing they are larger than previously known subgroups and answering an open question about their size.

Contribution

It introduces and analyzes the size of statistically characterized subgroups associated with continued fraction sequences, expanding understanding of their structure and answering an open problem.

Findings

Statistically characterized subgroups are strictly larger than characterized subgroups.

These subgroups have cardinality continuum and contain the subgroup generated by the irrational number.

The paper answers an open question about the size of these subgroups.

Abstract

In this note, we continue the investigation of the new version of characterized subgroups of the circle group $\mathbb{T}$, namely, "statistically characterized subgroups" (shortly, "s-characterized subgroups") recently introduced in \cite{DDB}. We primarily investigate these subgroups for sequences arising out of continued fraction representation of irrational numbers $\alpha$ in line of \cite{L} and \cite{KL} (followed by \cite{BDMW1}) comparing their main results for this new notion and show that these subgroups are strictly larger in size (so nontrivial) than the corresponding characterized subgroups, having cardinality $\mathfrak{c}$ and containing the subgroup $\langle \alpha \rangle$ and in the process answer the Open Question 6.4 posed in \cite{DDB}.