This is problem which came up in the process of designing a game.  Thus, I don't know any previous work relevant to the problem.

Fix a small set $D$ of a natural numbers.  For example, $D=\{1,2,3\}$.  Also fix natural number $k$, e.g., $k=5$.

We wish to consider a hitting set of all $k$-length arithmetic sequences $\{a, a+d, a+2d,...a+(k-1)d\}$ where $d \in D$ and $a > 0$.

Such a hitting set is a subset of the natural numbers $H$ such that for any $d \in D$ and $a > 0$, at least one of $a, a+d, ..., a+(k-1)d$ is contained in $H$

If the limit $\lim_{n \to \infty} |H \cap \{1,...,n\}|/n = \delta$ exists, then we say that $\delta$ is the density of the hitting set $H$.  For any given set $D$ and number $k$, there exists an infimal density $\delta_0$ of all such hitting sets.
I would like to estimate the infimal density $\delta_0$ for an arbitrary set $D$ and number $k$.

I give some concrete examples.

Let $D=\{1\}$.  Then the optimal hitting set is $C=\{k,2k,...\}$ with a density of $1/k$.

Let $D$ be a singleton $D=\{d\}$.  Then the optimal hitting set is $C=\{1,2,...,d-1,1+kd,2+kd,..,d-1+kd,1+2kd,...,d-1+2kd,...\}$ with a density of $1/k$.

Let $D=\{1,2\}$ and $k=2$.  Then an example of a hitting set is $C=\{n: n \mod 3 \in \{1,2\}\} = \{1,2,4,5,7,8,...\}$ with a density of 2/3.  However, I don't know whether a hitting set with a lower density exists. (EDIT: The answer is no, see comment by domotorp)

EDIT:

A visualization of the problem.  The top figure depicts the $k$-length arithmetic sequences for $k=6$ and $D=\{1,2,3\}$. hit by an element $n$, the red dot.  The black pixels in the first row are the values $a$ for which the $d=1$ sequences are hit by $n$.  The black pixels in the second row are the values $a$ for which the $d=2$ sequences are hit by $n$.  Analogously for the third row.
The figure on the bottom depicts a hitting set for the problem $k=6, D=\{1,2,3\}$ with a density $3/8$.

![enter image description here][1]


  [1]: https://i.sstatic.net/WcHhg.png