all possible combinations of $a_j$s for $\sum_{j=1}^na_j\epsilon^{j}=0$ where $\epsilon=e^{\frac{2\pi i}{n}}$

Question

Let $n\geq 3$ and $d$ be a divisor of $n$. Let us consider the set $\{1,\epsilon,\dots,\epsilon^{n-1}\}$ where $\epsilon=e^{\frac{2\pi i}{n}}$. Can we classify all possible cases for $d$ distinct elements from the set such that their sum is zero? I can see what is going on if we pick $d$ elements according to their exponents regarding modulo. I only consider the other cases that exponents for what we choose do not have to do with modulo.

Answer 1

(I use $\zeta$ for my $n$th root of unity, as is usual.)

If $d|n$, and $0\le a<d$, then the set $\{\zeta^a,\zeta^{a+d},\zeta^{a+3d},\ldots,\zeta^{a+n-d}\}$ does the job. We can call such a set $S^n_d(a)$.

But this doesn't give you all such sets. If we drop for a moment the requirement on the size of the set, we can have unions of two or more such sets, if they are disjoint; for instance, if $n=30$, we can have $A=\{1,\zeta^{10},\zeta^{20}\}\cup\{\zeta,\zeta^7,\zeta^{13},\zeta^{19},\zeta^{25}\}$.

But now if we consider these numbers as being embedded in the group of $120th$ roots of unity, then $d=8$ divides $120$, but $A$ is not of the form $S^{120}_8(a)$ for any $a$.