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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.

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  • $\begingroup$ Could you explicit a little bit what you need ? Why did you think of Dirac mass ? Because of the singular support ? You could have other examples of distributions like that. $\endgroup$
    – Ayman Moussa
    Commented Dec 5, 2020 at 22:19
  • $\begingroup$ Actually I've come to think there's probably no canonical way to do what I said. In the precise situation I was looking at, I realized that some additional information (essentially a set of generators) was playing a role and that was what gave meaning to the expression. $\endgroup$
    – alesia
    Commented Dec 5, 2020 at 22:41

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