Let $\mathbb{Z}$ be the set of integers, ${\mathbb{N}}_0$ the set of nonnegative integers and $F(x_1,x_2)=u_1{x}_1+u_2{x}_2$ be a binary linear form whose coefficients $u_1$, $u_2$ are nonzero, relatively prime integers such that $u_1{u}_2\ne \pm 1$ and $u_1{u}_2\ne -2$. Let $f:\mathbb{Z}\to {\mathbb{N}}_0\cup \left\{\infty \right\}$ be any function such that the set $f^{-1}\left(0\right)$ has asymptotic density zero. In 2007, M. B. Nathanson (2007) proved that there exists a set $A$ of integers such that $r_{A,F}\left(n\right)=f\left(n\right)$ for all integers $n$, where $r_{A,F}\left(n\right)=\left|\{(a,a^{\text{'}}):n=u_{1}a+u_2{a}^{\text{'}}:a,a^{\text{'}}\in A\}\right|$. We add the structure of difference for the binary linear form $F(x_1,x_2)$.

For any given positive integer k, and any set A of nonnegative integers, let $r_{1,k}(A,n)$ denote the number of solutions of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. We prove that if k,l are multiplicatively independent integers, i.e., log k/log l is irrational, then there does not exist any set A ⊆ ℕ such that both $r_{1,k}(A,n)=r_{1,k}(\mathbb{N}\setminus A,n)$ and $r_{1,l}(A,n)=r_{1,l}(\mathbb{N}\setminus A,n)$ hold for all n ≥ n₀. We also pose a conjecture and two problems for further research.