On the order of a minimal additive basis (Q1273711)

Let \(A\) be a set of non-negative integers. If every sufficiently large integer is the sum of \(h\) (not necessarily distinct) elements of \(A\), then \(A\) is said to be an asymptotic \(h\)-basis. If no proper subset of \(A\) is an asymptotic basis, then \(A\) is said to be minimal. \textit{P. Erdős} and \textit{M. B. Nathanson} [J. Number Theory 12, 154-159 (1980; Zbl 0426.10057)] proved that minimal asymptotic bases exist. They also asked if a set could be simultaneously an minimal asymptotic \(h\)-basis and a minimal asymptotic \(k\)-basis when \(h\) and \(k\) are integers with \(2\leq h < k\), and they proved that no such sets exist when \(h=2\) and \(k=4.\) In this article, the author gives a clever argument that generalizes the Erdős-Nathanson result; he proves that no such set can exist when \(h=2k\).