Some covering and packing results in number theory

AbstractCovering. Let Ak and B be sets of nonnegative integers consisting of k elements and infinitely many elements, respectively. The set B is called a complement of Ak if each nonnegative integer n is expressible in the form n = a + b (a ∈ A, b ∈ B). We define ck = supAkinfB d(B), where d(B) is the asymptotic density of B, B ranges over all complements of Ak, and Ak ranges over all sets consisting of k elements. Then 13 ≤ C4< 0.339934. Other covering results are proved. Packing. Let Ak and B be defined as above. The set B is called a packing complement of Ak if every nonnegative integer n has at most one representation n = a + b (a ∈ Ak, b ∈ B). We define Pk = supAkinfB d(B), where B ranges over all packing complements of Ak whose asymptotic density exists and Ak ranges over all k element sets. Then 1.(1) Pk≤(k2/4+11k/6−14)−1, k ⩾ 52.(2) 212≤limk→∞ k(Pk)12≤ lim supk→∞k(Pk)12≤8312