| Title:
|
Representation functions for binary linear forms (English) |
| Author:
|
Xue, Fang-Gang |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
74 |
| Issue:
|
1 |
| Year:
|
2024 |
| Pages:
|
301-304 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $F(x_1,x_2)$. (English) |
| Keyword:
|
representation function |
| Keyword:
|
binary linear form |
| Keyword:
|
density |
| MSC:
|
11B13 |
| MSC:
|
11B34 |
| idZBL:
|
Zbl 07893380 |
| idMR:
|
MR4717835 |
| DOI:
|
10.21136/CMJ.2024.0326-23 |
| . |
| Date available:
|
2024-03-13T10:10:52Z |
| Last updated:
|
2026-04-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152281 |
| . |
| Reference:
|
[1] Fang, J.-H.: Representation functions avoiding integers with density zero.Eur. J. Comb. 102 (2022), Article ID 103490, 7 pages. Zbl 1508.11015, MR 4350493, 10.1016/j.ejc.2021.103490 |
| Reference:
|
[2] Nathanson, M. B.: Representation functions of bases for binary linear forms.Funct. Approximatio, Comment. Math. 37 (2007), 341-350. Zbl 1146.11007, MR 2363831, 10.7169/facm/1229619658 |
| Reference:
|
[3] Xiong, R., Tang, M.: Unique representation bi-basis for the integers.Bull. Aust. Math. Soc. 89 (2014), 460-465 \99999DOI99999 10.1017/S0004972713000762 . Zbl 1301.11012, MR 3254755, 10.1017/S0004972713000762 |
| . |
