Weighted Representation Asymptotic Basis of Integers
Received:December 26, 2014  Revised:March 20, 2015
Key Word: additive basis   representation function
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11471017).
 Author Name Affiliation Yujie WANG School of Mathematics and Computer Science, Anhui Normal University, Anhui 241003, P. R. China Min TANG School of Mathematics and Computer Science, Anhui Normal University, Anhui 241003, P. R. China
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Let $k_{1}, k_{2}$ be nonzero integers with $(k_{1}, k_{2})=1$ and $k_{1}k_{2}\neq-1$. 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 $n\in\mathbb{Z}$.