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

Abstract

Let $h\geq2$ be an integer and let $1/h<\theta\leq1/(h-1)$. 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.