# Functions on dense subgroups of $\mathbb{R}^n$

Let $$G$$ be a finitely generated dense subgroup of $$\mathbb{R}^n$$, and $$f$$ be a character on $$G$$.

In the situation I'm looking at $$f$$ is either $$1$$ or $$-1$$ at any point.

Function $$f$$ can be extended to $$\mathbb{R}^n$$ by setting it to $$0$$ outside $$G$$. Is there any way to associate a distribution to $$f$$? For example one could try to form an infinite sum of Diracs on $$G$$, or perhaps a limit of suitably normalized finite sums of Diracs? Or maybe it can't be done in a meaningful way?

Any idea, reference or comment on how to deal with such objects is welcome.