整数集的加权表示的渐进基

Let k1, k2 be nonzero integers with(k1, k2) = 1 and k1k2≠-1. Let Rk1,k2(A, n)be the number of solutions of n = k1a1 + k2a2, where a1, a2 ∈ A. Recently, Xiong proved that there is a set A  Z such that Rk1,k2(A, n) = 1 for all n ∈ Z. Let f : Z-→ N0∪ {∞} 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  Z such that Rk1,k2(A, n) = f(n) for all n ∈ Z.

Weighted Representation Asymptotic Basis of Integers

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 $Asubseteqmathbb{Z}$ such that $R_{k_{1}, k_{2}}(A, n)=1$ for all $nin mathbb{Z}$. 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 $Asubseteq mathbb{Z}$ such that $R_{k_{1},k_{2}}(A, n)=f(n)$ for all $ninmathbb{Z}$.