Let $M\subset\mathbb{N}$ be a finite set. For every positive integer $n$ set $$D_n(M)=\{W\subset \mathbb{N} \text{ finite }|\ \forall\ i=0,\ldots,n-1\ \exists\ w\in W: w\in M+i\},$$ where $M+i=\{m+i|\ m\in M\}$. I would like to understand $$d_n(M)=Min\{|W|\ | \ W\in D_n(M)\}.$$ I am mainly interested in the asymptotic behaviour of the $d_n(M)$'s. So I would like to know $$d(M)=\lim_{n\to\infty}\frac{d_n(M)}{n}.$$

For example, for $M=\{1,2\}$ we have $(d_1(M),d_2(M),\ldots)=(1,1,2,2,3,3,\ldots)$ and so $d(M)=\frac{1}{2}$.

My question is, if this is (related to) a known combinatorial problem? It seems a fairly natural problem to me, so I could well imagine that it has been treated in the literature. I encountered it when trying to explicitly compute some value in an algebraic-geometric example.

I would like to know things like, is the sequence of first differences $\Delta d_n(M)=d_n(M)-d_{n-1}(M)$ always periodic?