Locally quasi-convex topologies on the group of the integers

Abstract

\noindent The most natural group topology on $\mathbb{Z}$ is the discrete one. There are other well-known group topologies on $\mathbb{Z}$, like the $p$-adic, defined for any prime number $p$. It is also an important group topology the weak topology with respect to the group of homomorphisms from $\mathbb{Z}$ to the unit circle of the complex plane; that is, the one defined by the characters and which is known as "the Bohr topology" on $\mathbb{Z}$. \noindent In \cite{tesislorenzo}, it is proved that taking as a neighbourhood basis at 0 the subsets $\{W_n\mid n\in \mathbb{N}\}$, defined by $W_n:=\{k\in \mathbb{Z}\mid \forall x\in S$, $k\cdot x\in [-\frac{1}{4n},\frac{1}{4n}]+\mathbb{Z}\}$, where $S$ is a quasi-convex sequence in $\T$, a group topology on $\mathbb{Z}$ is obtained, ${\tau }_S$. We know that the topology ${\tau }_S$ is metrizable and locally quasi-convex. \noindent In this monograph we characterize convergent sequences in ${\tau }_S$, for some $S\subset\T$. We give sufficient conditions on the elements of $\mathbb{Z}$ in order that they belong to a neighbourhood $W_n$. For a fixed sequence $(b_n)$ of natural numbers, restricted to mild conditions, we have considered the "linear topology associated" whose neighbourhood basis at 0 is $\{b_n\mathbb{Z}\mid n\in \mathbb{N}\}$ and the "topology of uniform convergence" on $S=\{\frac{1}{b_n}+\Z\mid n\in\N\}\subset\T$. We make a comparative study between both classes of topologies.

arXiv comment53 pages