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For $n \geq 3$, say that a centered $n$-gon with $L$ layers is given by the origin, $(0,0)$ together with the points $$ \left\{\alpha\zeta_n^j + \beta\zeta_n^{j+1}\ \middle\vert\ 0 \leq j < n, 1 \leq \alpha \leq L, 0 \leq \beta \leq L - \alpha\right\}, $$ where $\zeta_n$ is a primitive $n$th root of unity.

For example, here are the cases when $(n,L) = (3,4)$, $(n,L) = (4,4)$, $(n,L) = (5,3)$, and $(n,L) = (6,3)$:


Question

For a given integer $k \geq 3$, what are the conditions on $n$ and $L$ such that it is possible to find $k$ points that form a regular $k$-gon in the centered $n$-gon with $L$ layers?

Besides the hexagon in the centered $3$-gon, are there any examples of $k$-gons in a centered $n$-gon where $k\nmid n$?

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