Weighted Representation Asymptotic Basis of Integers 
Received:December 26, 2014 Revised:March 20, 2015 
Key Words:
additive basis representation function

Fund Project:Supported by the National Natural Science Foundation of China (Grant No.11471017). 

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Abstract: 
Let $k_{1}, k_{2}$ be nonzero integers with $(k_{1}, k_{2})=1$ and $k_{1}k_{2}\neq1$. Let $R_{k_{1}, k_{2}}(A, n)$ be the number of solutions of $n=k_{1}a_{1}+k_{2}a_{2}$, where $a_{1}, a_{2}\in A$. Recently, Xiong proved that there is a set $A\subseteq\mathbb{Z}$ such that $R_{k_{1}, k_{2}}(A, n)=1$ for all $n\in \mathbb{Z}$. Let $f: \mathbb{Z}\longrightarrow \mathbb{N}_{0}\cup\{\infty\}$ be a function such that $f^{1}(0)$ is finite. In this paper, we generalize Xiong's result and prove that there exist uncountably many sets $A\subseteq \mathbb{Z}$ such that $R_{k_{1},k_{2}}(A, n)=f(n)$ for all $n\in\mathbb{Z}$. 
Citation: 
DOI:10.3770/j.issn:20952651.2015.06.012 
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