On Minimal Asymptotic Basis of Order 4
Received:April 27, 2016  Revised:September 07, 2016
Key Word: minimal asymptotic basis   $g$-adic representation
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11471017).
 Author Name Affiliation Jingwen LI School of Mathematics and Computer Science, Anhui Normal University, Anhui 241003, P. R. China Jiawen LI School of Mathematics and Computer Science, Anhui Normal University, Anhui 241003, P. R. China
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Let $\mathbb{N}$ denote the set of all nonnegative integers and $A$ be a subset of $\mathbb{N}$. Let $W$ be a nonempty subset of $\mathbb{N}$. Denote by $\mathcal{F}^{\ast}(W)$ the set of all finite, nonempty subsets of $W$. Fix integer $g\geq2$, 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 $i=0,1,2,3$, let $W_{i}=\{n\in \mathbb{N} \mid n\equiv i \pmod 4\}$. In this paper, we show that the set $A=\bigcup_{i=0}^3 A_{g}(W_{i})$ is a minimal asymptotic basis of order four.