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}.

中文翻译：

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使用广义的密度函数类生成圆的子群

在本文中，我们考虑在 \cite{BDK} 中引入的自然密度函数的广义版本 $d^f_g$ 其中 $g : \N \rightarrow [0,\infty)$ 满足 $g(n) \rightarrow \ infty$ 和 $\frac{n}{g(n)} \nrightarrow 0$ 而 $f$ 是一个无界模函数，并使用这些密度函数生成圆群 $\T$ 的特征子群的版本。我们表明这些子群与 $s$ 特征化子群 \cite{DDB} 或 $\alpha$ 特征化子群 \cite{BDH} 具有相同的特征，我们的结果提供了两篇文章主要结果的更一般版本. 但同时，这种更通用的方法的效用是通过为 $f$ 和 $g$ 的合适选择构造新的和非平凡的子群来证明的。