On asymptotic bases and minimal asymptotic bases
Volume 170 / 2022
Colloquium Mathematicum 170 (2022), 65-77
MSC: Primary 11B13.
DOI: 10.4064/cm8321-9-2021
Published online: 20 April 2022
Abstract
Let $\mathbb {N}=\{0,1,2,\ldots \}$ and $A\subset \mathbb {N}$. Let $h\geq 2$ and let $r_h(A,n)=\sharp \{ (a_1,\ldots ,a_h) \in A^{h}: a_1+\cdots +a_h=n\}.$ The set $A$ is called an asymptotic basis of order $h$ if $r_h(A,n)\geq 1$ 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 resolved a problem of Nathanson on minimal asymptotic bases of order $h$. In this paper, we generalize this result to $g$-adic representations.
