One defines a subset \(\Lambda\subseteq\mathbb{R}\) to be uniformly discrete (u.d.) if there is an \(\varepsilon>0\) such that each pair in \(\Lambda\) is at least \(\varepsilon\) apart. For such \(\Lambda\) there is a natural density concept (lower uniform) \(d(\Lambda)\leq 1/\varepsilon\). In this article, we study the following problem defined by means of this density concept. Fix a set \(A\subseteq\mathbb{R}\), \(A\neq\emptyset\). Find a u.d. set \(\Lambda\), of minimal density which meets all the integral translates of \(A\), i.e. \(A\), \(A+1,A-1,A+2,\dots\)\ . It is proved that such a set exists and can be chosen to be periodic, i.e., the union of finitely many bothways infinite arithmetic sequences, all with the same difference. The last part of the article contains partial results for determining \(\Lambda\) for certain finite sets \(A\). This is a very difficult combinatorial problem.