Representation functions for binary linear forms

Xue, Fang-Gang. "Representation functions for binary linear forms." Czechoslovak Mathematical Journal (2024): 301-304. <http://eudml.org/doc/299235>.

@article{Xue2024,
abstract = {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 $u_1u_2\ne \pm 1$ and $u_1u_2\ne -2$. Let $f\colon \mathbb \{Z\}\rightarrow \mathbb \{N\}_0\cup \lbrace \infty \rbrace $ 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 $r_\{A,F\}(n)=|\lbrace (a,a^\{\prime \}) \colon n=u_1a+u_2a^\{\prime \} \colon a,a^\{\prime \}\in A\rbrace |$. We add the structure of difference for the binary linear form $F(x_1,x_2)$.},
author = {Xue, Fang-Gang},
journal = {Czechoslovak Mathematical Journal},
keywords = {representation function; binary linear form; density},
language = {eng},
number = {1},
pages = {301-304},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Representation functions for binary linear forms},
url = {http://eudml.org/doc/299235},
year = {2024},
}

TY - JOUR
AU - Xue, Fang-Gang
TI - Representation functions for binary linear forms
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 1
SP - 301
EP - 304
AB - 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 $u_1u_2\ne \pm 1$ and $u_1u_2\ne -2$. Let $f\colon \mathbb {Z}\rightarrow \mathbb {N}_0\cup \lbrace \infty \rbrace $ 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 $r_{A,F}(n)=|\lbrace (a,a^{\prime }) \colon n=u_1a+u_2a^{\prime } \colon a,a^{\prime }\in A\rbrace |$. We add the structure of difference for the binary linear form $F(x_1,x_2)$.
LA - eng
KW - representation function; binary linear form; density
UR - http://eudml.org/doc/299235
ER -