Let $A$, $B$ be two finite subsets of integers. We denote by $C(A, B)$ the minimum number of shifts of $A$ to cover $B$. More formally, it can be written as
$$ C(A, B)=\min\{|S|: S\subseteq B-A,~\text{such that}~B\subseteq A+S\}. $$
The trivial lower bound for $C(A, B)$ is $|B|/|A|$. Random simulations show the following upper bound
$$ C(A, B)\leq \frac{|B-A|}{|A|}. $$
I googled covering number stuff, but do not see this upper bound. Inductive argument should work, but I can not prove it. Anybody knows this result or ideas to prove it?