Additive bases and subadditive functions (Q1288891)

A sequence \(A=\{a_1,a_2,\dots, a_n,\dots\}\) of non-negative integers is called an additive basis if there exists an \(h\) such that any non-negative integer can be expressed as a sum of \(h\) terms of \(A\). The author proves the following result: If the function \(f\) is non-negative bounded subadditive (i.e. \(0\leq f(m+n)\leq f(m)+f(n)\)) then either \(\lim_{a\in A, a\to\infty} f(a)=0\) implies that \(A\) is not a basis or there exists a positive integer \(k\) such that \(\lim_{n\to\infty}f(kn)=0\). This result is connected with the paper of \textit{J. M. Deshouillers, P. Erdős} and \textit{A.Sárközy} [Acta Arith. 30, 121-132 (1976; Zbl 0349.10047)].