  In this article, we consider the generalized version $d^f_g$ of the natural
density function introduced in \cite{BDK} where $g : \N \rightarrow [0,\infty)$
satisfies $g(n) \rightarrow \infty$ and $\frac{n}{g(n)} \nrightarrow 0$ whereas
$f$ is an unbounded modulus function and generate versions of characterized
subgroups of the circle group $\T$ using these density functions. We show that
these subgroups have the same feature as the $s$-characterized subgroups
\cite{DDB} or $\alpha$-characterized subgroups \cite{BDH} and our results
provide more general versions of the main results of both the articles. But at
the same time the utility of this more general approach is justified by
constructing new and nontrivial subgroups for suitable choice of $f$ and $g$.
In several of our results we use properties of the ideal $\iZ_g(f)$ which are
first presented along with certain new observations about these ideals which
were not there in \cite{BDK}.
