Geodesic
Representation functions for binary linear forms
Czechoslovak Mathematical Journal, Tome 74 (2024) no. 1, pp. 301-304
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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 $F(x_1,x_2)$.
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 $F(x_1,x_2)$.
DOI : 10.21136/CMJ.2024.0326-23
Classification : 11B13, 11B34
Keywords: representation function; binary linear form; density 
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Xue, Fang-Gang. Representation functions for binary linear forms. Czechoslovak Mathematical Journal, Tome 74 (2024) no. 1, pp. 301-304. doi: 10.21136/CMJ.2024.0326-23

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