This is a follow up to my previous question (see linked question).
In short, there it is shown that if $n$ is prime, then any set can make it.
I want to characterize sets $\mathbb A_n$ of multisets $A$, such that the following claim holds:
Let $n\gt 2$. For any multiset $A\not\in \mathbb A_n$ of $n-1$ non-zero elements of $\mathbb Z/n\mathbb Z$, we can make zero using $+,-$.
Problem is finding all $A$ such that $A\in\mathbb A_n$.
Due to Cauchy-Davenport theorem, we know that: For any $n-1$ non-zero elements of $\mathbb Z/n\mathbb Z$, we can make zero using $+,-$ if and only if $n$ is prime. Therefore, if $n$ is prime then $\mathbb A_n=\emptyset$.
It remains to consider composite $n$.
a) Let $n$ be odd composite. Let $p$ be a fixed prime factor of $n$.
It seems that if $A$ consists of $n-2$ multiples of $p$ and one non-multiple of $p$, then $A\in\mathbb A_n$.
What other multisets $A$ satisfy $A\in\mathbb A_n$? I do not know of any other examples.
b) Let $n$ be even composite.
If $A$ contains an odd number of odd integers, then clearly $A\in\mathbb A_n$.
What other multisets are also in $\mathbb A_n$?
Number of other multisets seems to be $1,2,27,32,1000\dots$ for $n=4,6,8,10,12,\dots$
For example, for $n=4$ and $n=6$ they are $\{2,2,2\}$ and $\{3, 3, 3, 3, 2\},\{3, 3, 3, 3, 4\}$.
