Weighted Representation Asymptotic Basis of Integers

DOI：10.3770/j.issn:2095-2651.2015.06.012

 作者 单位 王玉杰 安徽师范大学数学计算机科学学院, 安徽 芜湖 241003 汤敏 安徽师范大学数学计算机科学学院, 安徽 芜湖 241003

令$k_{1}, k_{2}$ 是满足$(k_{1}, k_{2})=1$且$k_{1}k_{2}\neq-1$的非零整数. 令$R_{k_{1}, k_{2}}(A, n)$表示$n=k_{1}a_{1}+k_{2}a_{2}$的解的个数, 其中$a_{1}, a_{2}\in A$.最近, 熊然证明了存在集合$A\subseteq\mathbb{Z}$使得每个整数$n$都可以唯一地表示成$n=k_{1}a_{1}+ k_{2}a_{2}$. 令$f: \mathbb{Z}\longrightarrow \mathbb{N}_{0}\cup\{\infty\}$ 是满足$f^{-1}(0)<\infty$的函数. 在本文中, 我们推广熊然的结论，证明了存在集合$A\subseteq \mathbb{Z}$ 使得每个整数$n$都满足$R_{k_{1},k_{2}}(A, n)=f(n)$.

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 $n\in\mathbb{Z}$.