On asymptotic bases and minimal asymptotic bases

Abstract

Let $\mathbb{N}$ denote the set of all nonnegative integers and $A$ be a subset of $\mathbb{N}$. Let $h\ge 2$ and let $r_h(A,n)=♯\{(a_1,\dots ,a_h)\in A^h:a_1+\cdots +a_h=n\}.$ The set $A$ is called an asymptotic basis of order $h$ if $r_h(A,n)\ge 1$ for all sufficiently large integers $n$. An asymptotic basis $A$ of order $h$ is minimal if no proper subset of $A$ is an asymptotic basis of order $h$. Recently, Chen and Tang resoved a problem of Nathanson on minimal asymptotic bases of order $h$. In this paper, we generalized this result to $g$-adic representations.