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Daniele Impieri

Publications and source records attributed to Daniele Impieri.

3 recordsLinked to original sources
arXiv2015

Characterized subgroups of topological abelian groups

A subgroup $H$ of a topological abelian group $X$ is said to be characterized by a sequence $\mathbf v =(v_n)$ of characters of $X$ if $H=\{x\in X:v_n(x)\to 0\ \text{in}\ \mathbb T\}$. We study the basic properties of characterized subgroups in the general setting, extending results known in the compact case. For a better description, we isolate various types of characterized subgroups. Moreover, we introduce the relevant class of autochacaracterized groups (namely, the groups that are characterized subgroups of themselves by means of a sequence of non-null characters); in the case of locally compact abelian groups, these are proved to be exactly the non-compact ones. As a by-product of our results, we find a complete description of the characterized subgroups of discrete abelian groups.

math.GN↗
arXiv2015

On the Borel Complexity of Characterized Subgroups

In a compact abelian group $X$, a characterized subgroup is a subgroup $H$ such that there exists a sequence of characters $\vs=(v_n)$ of $X$ such that $H=\{x\in X:v_n(x)\to 0 \text{ in } \T\}$. Gabriyelyan proved for $X=\T$, that $\{x\in\T:n!x\to 0 \text{ in }\T\}$ is not an $F_σ$-set. In this paper, we give a complete description of the $F_σ$-subgroups of $\T$ characterized by sequences of integers $\vs=(v_n)$ such that $v_n|v_{n+1}$ for all $n\in\N$ (we show that these are exactly the countable characterized subgroups). Moreover in the general setting of compact metrizable abelian groups, we give a new point of view to study the Borel complexity of characterized subgroups in terms of appropriate test-topologies in the whole group.

math.GN↗