number of sums in $\mathbb{Z}_{p^r}$ which are coprime to $p^r$

Question

We look at the ring of integers modulo a prime power, say $p^r$ and $r>1$. Eulers totient formula says that there are $p^r-p^{r-1}$ elements in this ring $\mathbb{Z}_{p^r}$ that are coprime to $p^r$. Denote this set of coprime numbers by $C\subset \mathbb{Z}_{p^r}$. Suppose now $x\in C$. I am curious about the cardinality of the set $C_x:=\{y\in C: x+y\in C\}$. Or written differently, $C_x=\{z\in x+C: z\in C\}=(x+C)\cap C$. Writing it in this way, makes me think if there are results in additive combinatorics which say something about $|C_x|$. If someone knows results which apply to this case, please let me know. Thanks.

Answer 1

If we fix $$x\ne 0\mod p$$ then we have two conditions for $y$ :

$$y\ne 0\mod p$$ $$y\ne -x\mod p$$

Hence , of the $p$ possible residues , $y$ is allowed to have $p-2$. Hence, there are $$\frac{p-2}{p}\cdot p^r=(p-2)\cdot p^{r-1}$$ numbers in the range $[0,p^r-1]$ with the desired property.