On Minimal Asymptotic Basis of Order 4 |
Received:April 27, 2016 Revised:September 07, 2016 |
Key Words:
minimal asymptotic basis $g$-adic representation
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Fund Project:Supported by the National Natural Science Foundation of China (Grant No.11471017). |
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Abstract: |
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. |
Citation: |
DOI:10.3770/j.issn:2095-2651.2016.06.003 |
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