  \noindent The most natural group topology on $\Z$ is the discrete one. There
are other well-known group topologies on $\Z$, 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 $\Z$ 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 $\Z$.
  \noindent In \cite{tesislorenzo}, it is proved that taking as a neighbourhood
basis at 0 the subsets $\{W_n\mid n\in\N\}$, defined by
$W_n:=\{k\in\Z\mid\forall x\in S$, $ k\cdot
x\in[-\frac{1}{4n},\frac{1}{4n}]+\Z\}$, where $S$ is a quasi-convex sequence in
$\T$, a group topology on $\Z$ 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 $\Z$ 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 $\{b_n\Z\mid
n\in\N\}$ 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.
