Derived Set of a given subset of Real Line. Let $A = \{a +\pi b : a , b \in \mathbb{Z}\}$ is a subset of $\mathbb{R}$. What will be the derived set of it?

$A$ is an additive subgroup of $\Bbb R$, which is not discrete since $\pi$ is irrational. A known result about additive sub-groups of $\Bbb R$ shows that $A$ is dense in $\Bbb R$ for the usual topology. Each point which is not in $A$ is a limit point. 
For the points which are in $A$, we use irrationality criteria (proposition 4 in the link).
