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In a comment to my recent question about covering segments by a disk, Gerhard Paseman has suggested a generalisation: replacing the segments of the original $n$-gon by a simple closed (say, convex) ...

Given two integers $n$ and $N$ such that $N>{n+1\choose 2}$, we denote by $\alpha_n(N)$ the number of elements $(x_1,\dots,x_n)$ in $(\mathbb Z/N\mathbb Z)^n$ such that the $2n$ elements $x_1,x_1+1,...