On a theorem of Plünnecke concerning the sum of a basis and a set of positive density (Q1900860)

In 1970 \textit{H. Plünnecke} [J. Reine Angew. Math. 243, 171-183 (1970; Zbl 0199.36701)] developed a graph-theoretic method in additive number theory and used it to prove the inequality \((*)\) \(\sigma (A + B) \geq \sigma (A)^{1 -1/h}\), where \(\sigma\) denotes Schnirelmann density, \(A\) is any set and \(B\) is any basis of order \(h\). In spite of its sharpness and the wide applicability of the method, this remained relatively unknown for a long time. The reviewer [Sci., Ser. A 3, 97-109 (1989; Zbl 0743.05052)] simplified this proof and gave further applications. The present paper gives a very clear exposition of a complete proof of \((*)\). The graph-theoretic part follows the reviewer's paper with certain changes, among others, a definitely better terminology. The derivation of \((*)\) from graph theory simplifies Plünnecke's proof.