Statistically characterized subgroups related to arithmetic-type sequence of integers

TL;DR

This paper extends the study of statistically characterized subgroups to a broader class of arithmetic-type sequences, revealing new behaviors and unifying previous results for arithmetic and certain non-arithmetic sequences.

Contribution

It introduces a generalized framework for statistically characterized subgroups that encompasses previous special cases and demonstrates their distinct behaviors.

Findings

Previous cardinality results are special cases of the new framework.

The broader class exhibits significantly different behavior from known cases.

Unified results for arithmetic and some non-arithmetic sequences are established.

Abstract

Very recently, in [Das et al., J. Lond. Math. Soc., 2025], statistically characterized subgroups were studied for certain classes of non-arithmetic sequences. Subsequently, in [Das et al., Bull. Sci. Math., 2025], characterized subgroups were investigated for a class of arithmetic-type sequences that includes both arithmetic sequences and certain non-arithmetic sequences. Motivated by these developments, we study statistically characterized subgroups associated with a broader class of arithmetic-type sequences. In particular, all previously obtained cardinality related observations for statistically characterized subgroups corresponding to arithmetic sequences as well as certain non-arithmetic sequences follow as special cases of our results. Moreover, we show that this broader class exhibits drastically different behavior and differs significantly from the previously studied special cases.