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On minimal asymptotic bases of order three

Volume 151 / 2018

Dengrong Ling, Min Tang 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.

Authors

  • Dengrong LingSchool of Mathematics and Computer Science
    Anhui Normal University
    Wuhu 241003, P.R. China
    e-mail
  • Min TangSchool of Mathematics and Computer Science
    Anhui Normal University
    Wuhu 241003, P.R. China
    e-mail

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