On a problem of Nathanson related to strongly minimal asymptotic bases of order $h$

Let $h\ge 2$ be an integer and let $1\mathrm{/}h<\theta \le 1\mathrm{/}(h-1)$. In this paper, we prove that there exists a strongly minimal asymptotic basis $A$ of order $h$ such that $A(x)\asymp x^{\theta }$. This solves a problem posed by Nathanson in 1988.