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Keywords:
representation function; binary linear form; density
representation function; binary linear form; density
Summary:
Let $\mathbb {Z}$ be the set of integers, $\mathbb {N}_0$ the set of nonnegative integers and $F(x_1,x_2)=u_1x_1+u_2x_2$ be a binary linear form whose coefficients $u_1$, $u_2$ are nonzero, relatively prime integers such that $u_1u_2\neq \pm 1$ and $u_1u_2\neq -2$. Let $f\colon \mathbb {Z}\rightarrow \mathbb {N}_0\cup \{\infty \}$ 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 $r_{A,F}(n)=|\{(a,a') \colon n=u_1a+u_2a' \colon a,a'\in A\}|$. We add the structure of difference for the binary linear form $F(x_1,x_2)$.
References:
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[2] Nathanson, M. B.: Representation functions of bases for binary linear forms. Funct. Approximatio, Comment. Math. 37 (2007), 341-350. DOI 10.7169/facm/1229619658 | MR 2363831 | Zbl 1146.11007
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