Article
| 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 |
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| Category: | math |
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| 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 |
| DOI: | 10.21136/CMJ.2024.0326-23 |
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| Date available: | 2024-03-13T10:10:52Z |
| Last updated: | 2024-03-18 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152281 |
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| 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 |
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