What is the measure of the sets of perfect nth powers of p-adic units? Specifically I'm working with the Haar measure normalized to $\mu(\mathbb{Z}_p)=1$ on the p-adic integers. More precisely, I define the sets of perfect nth powers of units as,
$$S_n = \{u | u= z^n, z \in \mathbb{Z}_p^\times\}$$
and I'm wanting to evaluate, $$\mu(S_n)$$
That's the end of my question, however here are four results I was able to at least figure out on my own that I'm still thinking about to help me whittle it down and understand further.
Since $S_1$ is just the entire set of units, $$\mu(S_1) = 1- \frac{1}{p}$$ An upper bound can be found if d divides n, then $S_n \subseteq S_d$, so
$$\mu(S_n) \le \mu(S_d)$$
and further using Euler's theorem I figured out that in $S_{\varphi(p^k)}$ since $u=z^{\varphi(p^k)}\equiv 1 \mod p^k$ then $u \in 1 + p^k \mathbb{Z}_p$ so it must be $S_{\varphi(p^k)} \subseteq 1+p^k\mathbb{Z}_p$ and,
$$\mu(S_{\varphi(p^k)}) \le \frac{1}{p^k}$$
If $m$ divides $p-1$, then $\mathbb{Z}_p$ contains a $m$th root of 1 and the elements of $S_m$ can be made by only picking elements from $\frac{p-1}{m}$ of the subsets of $\mathbb{Z}_p^\times$ of the form $a+p\mathbb{Z}_p$ and raising them to the $m$th power, so
$$\mu(S_m) \le \frac{p-1}{m p}$$