For a finite set S of integers, a packing set is a set T such that the translates S+t (t in T) are pairwise disjoint, and the packing density dp(S) is the supremum of the natural upper densities of such T. Every 4-element integer set turns out to admit a packing set of natural density at least 1/7, and 1/7 is best possible: the set {0,1,4,6} attains it exactly. Establishing this requires reducing the problem to finitely many cases via periodicity of an optimal packing modulo the diameter of S, a pigeonhole argument that any packing can be replaced by an eventually periodic one of no smaller density, and a greedy construction that certifies the lower bound for every remaining set. The result sharpens the known general lower bound for sets of small cardinality and separates the four-element case from the easier two- and three-element cases, where the extremal densities are 1/2 and 1/3.
We prove that the set $\{0, 1, 4, 6\}$ achieves the minimum packing density among all sets of integers with cardinality four, with a density of $\frac{1}{7}$.