On minimal asymptotic bases of order three
Volume 151 / 2018
Colloquium Mathematicum 151 (2018), 9-18
MSC: Primary 11B13.
DOI: 10.4064/cm6901-2-2017
Published online: 13 October 2017
Abstract
Let $A$ be a subset of $\mathbb {N}$, and $W$ be a nonempty subset of $\mathbb {N}$. Denote by $\mathcal {F}^{\ast }(W)$ the set of all finite, nonempty subsets of $W$. For integer $g\geq 2$, let $A_{g}(W)$ be the set of all numbers of the form $ \sum _{f\in F}a_{f}g^{f}$ where $F\in \mathcal {F}^{\ast }(W)$ and $1\leq a_{f}\leq g-1$. For $i=0,1,2$, let $W_{i}=\{n\in \mathbb {N} \mid n\equiv i ({\rm mod} 3)\}$. We show that for any $g\geq 2$, the set $A=A_{g}(W_{0})\cup A_{g}(W_{1})\cup A_{g}(W_{2})$ is a minimal asymptotic basis of order three. Moreover, we construct an asymptotic basis of order three containing no subset which is a minimal asymptotic basis of order three.
