We introduce a new version of characterized subgroups of the circle group $$\mathbb{T}$$ that we call " $$\alpha$$ -statistically characterized subgroups", in line of the very recent work [11], using the notion of statistical convergence of order $$\alpha$$ [4]. We show that for any arithmetic sequence $$(a_n)$$ and $$\alpha \in (0, 1)$$ , the $$\alpha$$ -statistically characterized subgroups $$t^\alpha_{(a_n)}(\mathbb{T}) $$ is again a Borel subgroup having cardinality $$\mathfrak{c}$$ lying between the characterized and statistically characterized subgroups.

中文翻译：

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使用 α 阶统计收敛生成圆的子群

我们介绍了圆群 $$\mathbb{T}$$ 的新版本特征子群，我们称之为“$$\alpha$$ - 统计特征子群”，符合最近的工作 [11]，使用$$\alpha$$ [4] 阶统计收敛的概念。我们证明，对于任何等差数列 $$(a_n)$$ 和 $$\alpha \in (0, 1)$$ ，$$\alpha$$ 统计表征的子群 $$t^\alpha_{(a_n) }(\mathbb{T}) $$ 又是一个 Borel 子群，其基数 $$\mathfrak{c}$$ 位于特征化和统计特征化的子群之间。