Let $n\geq 2$ be a positive integer and $k$ be a number between $1$ and $n$. Recently, I came across the following question about $\mathbb Z/n\mathbb Z$ and I wonder if it was studied before. I'd be thankful for any references or any suggestions on how to approach the problem.


Find the smallest integer $t$ (or an upper bound on $t$) such that, no matter which $t$ distinct *non-zero* elements $i\_1,i\_2,\ldots,i\_t \in \mathbb Z/n\mathbb Z$ you pick, each element $j\in \mathbb Z/n\mathbb Z$ can be written in the form


$$
j \equiv \ell i\_r \pmod n,
$$


where $0\leq \ell \leq k$ and $1\leq r \leq t$. In other words, $t$ $(k+1)$-term arithmetic progressions of the form $0, i\_r, 2i\_r, \ldots, ki\_r$ entirely cover $\mathbb Z/n\mathbb Z$.


I am especially interested in the case when $n$ is prime and $k \approx n/2$.



Covering a finite ring with arithmetic progressions
The case of prime $n$. 


We may suppose that $j\ne 0$. Consider $t$ distinct elements $j/i\_1,\dots,j/{i\_{t}}$. If all of them belong to the set $\{k+1,\dots,n-1\}$, we get $t\leqslant n-k-1$. Thus for $t=n-k$ we may always find $r$ such that $j/i\_r\in \{1,2,\dots,k\}$ as you need. For $t=n-k-1$ this is not always possible: take $j=1$, $i\_r=1/(n-r)$ for $r=1,\dots,n-k-1$.


