Representation functions for binary linear forms

Let Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{Z}$$\end{document} be the set of integers, N0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{N}_{0}$$\end{document} 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:Z→N0∪{∞}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f : \mathbb{Z}\rightarrow \mathbb{N}_{0}\ \cup\{\infty\}$$\end{document} 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 rA,F(n) = f(n) for all integers n, where rA,F(n) = ∣{(a,a′): n = u1a + u2a′: a, a′ ∈ A}∣. We add the structure of difference for the binary linear form F(x1,x2).