Representation functions for binary linear forms

Let \(\mathbb{Z}\) $$\mathbb{Z}$$ be the set of integers, \(\mathbb{N}_{0}\) $${\mathbb{N}}_0$$ the set of nonnegative integers and F(x₁,x₂) = u₁x₁ + u₂x₂ be a binary linear form whose coefficients u₁, u₂ are nonzero, relatively prime integers such that u₁u₂ ≠ ±1 and u₁u₂ ≠ −2. Let \(f : \mathbb{Z}\rightarrow \mathbb{N}_{0}\ \cup\{\infty\}\) $$f:\mathbb{Z}\to {\mathbb{N}}_0\cup \{\mathrm{\infty }\}$$ be any function such that the set f⁻¹(0) has asymptotic density zero. In 2007, M. B. Nathanson (2007) proved that there exists a set A of integers such that r_A,F(n) = f(n) for all integers n, where r_A,F(n) = ∣{(a,a′): n = u₁a + u₂a′: a, a′ ∈ A}∣. We add the structure of difference for the binary linear form F(x₁,x₂).