Let $\mathbb{Z}_n$ be the set of integers in modulo $n$: $\{0,1,...,n-1\}$. Note that $n$ does not have to be prime. And $S_m$ be the subset of $\mathbb{Z}_n$, containing $m$ different elements in $\mathbb{Z}_n$: $\{i_1,...,i_m\}$. We only know the number $m$, but not the set $S_m$ itself.
Then I want to have $\mathbb{Z}_n$ by adding and subtracting $S_m$: is it possible to say that for some $m$, $\mathbb{Z}_n=S_m\pm S_m$ where $\pm$ is (Minkowski) addition and subtraction as a set in modulo $n$? Then what is the condition for $m$? (I am guessing $m\geq n/2$ would work, but not sure how to prove.)
Further, the same question for $\mathbb{Z}_n=(S_m\pm S_m)\pm S_m$, to have this, what is the conditon for $m$?
If needed, we can assume that $S_m$ contains $0$ and $1$ for $m\geq 2$, but I don't think this matters... [EDIT: as indicated in the comment, having $0$ and $1$ matters. It may help to have small $m$, so please assume.]
Any comment is appreciated!