Consider the field of $p$-adic numbers $\mathbb Q_p$. Define the character $\chi(u p^n) = e(p^n)$ for all $n \in \mathbb Z$ and all unit $u$. In particular it is trivial on integers. This allows to define a Fourier transform and I am interested in computing the one of $\mathbb{1}_{\mathcal O_p}$ (characteristic function of the $p$-adic integers $\mathcal O_p$.
I tried to cut by valuations, without success (the measure is normalized so that $\mathcal O_p$ gets measure one). Let $y = p^{-n} v$ for $n \in \mathbb Z$ and $v$ a unit: \begin{align} \int_{\mathbb Q_p} e_p(-x y) \mathbb{1}_{\mathcal O_p}(x) dx & = \int_{\mathcal O_p} e_p(-x y) dx \\ & = \sum_{k \geq n} \int_{\mathcal p^k O_p^\times} e_p(-x y) dx + \sum_{0\leq k < n} \int_{\mathcal p^k O_p^\times} e_p(-x y) dx \end{align}
Now I can write $x = p^k u \in p^k \mathcal{O}_p^\times$, and get \begin{align} & \sum_{k \geq n} \int_{\mathcal O_p^\times} e_p(-p^{k-n} uv) p^{-k} du + \sum_{0 \leq k < n} \int_{\mathcal p^k O_p^\times} e_p(-p^{k-n} uv) p^{-k} du \\ & = \sum_{k \geq n} p^{-k} \mathrm{vol}(\mathcal O_p^\times) + \sum_{0 \leq k < n} e(p^{k-n}) p^{-k} \mathrm{vol}(\mathcal O_p^\times) \end{align}
When $n \leq 0$, i.e. $y$ is integer, I indeed get 1 as expected. However when it is not the case I don't know how to compute this sum (and, expectedly, get 0).
For $a,b\in \Bbb{Z}_p$ and $k\ge 0$ an integer $$\exp(-2i\pi ab/p^k)=\exp(-2i\pi (ab\bmod p^k)/p^k)$$ is well-defined. Then
$$\int_{\mathbb Z_p} \exp(-2i\pi x y)dx=\lim_{n\to \infty}\frac1{p^n} \sum_{a=0}^{p^n -1} \exp(-2i\pi a y)= 1_{y\in \Bbb{Z}_p}$$