Given a regular $n$-gon ($n \geq 3$ integer), consider $t$ subsets $S_1,..,S_t$ of the vertices. Depending on $n, t$, and on the size of the sets $S_i$ one may find rotations $r_i , i = 1 .. t$ of the $n$-gon such that $r_i(S_i) \cap r_j(S_j) = \emptyset$ for all $i \neq j$ in $1 .. t$. In this case we say that the sets can be rotated apart.
In my paper "Shuffled equi-$n$-squares" available at
http://arxiv.org/abs/1701.02325
I proved that if $n = (t-1)*m^2$ with $t \geq 3$ and $m \geq 2$, then any $t$ sets of size $m$ can be rotated apart in a regular $n$-gon. The proof uses cyclotomic polynomials. As to the sharpness of this result I have a
Question. Given $n = 2*m^2 - 1$ with $m \geq 2$, is it true that every three sets of size $m$ can be rotated apart?
My special interest goes to a regular 127-gon and three sets of size $8$. A negative answer to the Question with $n = 127$ would also confirm the sharpness of another, general, argument proving that $t$ sets can be rotated apart in a regular $n$-gon if the sum $s$ of the sizes satisfies $(t - 1)*s^2/t^2 < n$ (lemma 3.11). Any answer is welcome, however.
Note: The question has been reformulated in a more general way to gain clarity.