Correct order on some certain weighted representation functions
Let $\mathbb{N}$ be the set of all nonnegative integers. For any positive integer $k$ and any subset $A$ of nonnegative integers, let $r_{1,k}(A,n)$ be the number of solutions $(a_1,a_2)$ to the equation $n=a_1+ka_2$. In 2016, Qu proved that \[ \liminf _{n\,\rightarrow \,\infty }r_{1,k}(A,n)=\infty...
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Académie des sciences
2024-05-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.573/ |
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| author | Chen, Shi-Qiang Ding, Yuchen Lü, Xiaodong Zhang, Yuhan |
| author_facet | Chen, Shi-Qiang Ding, Yuchen Lü, Xiaodong Zhang, Yuhan |
| author_sort | Chen, Shi-Qiang |
| collection | DOAJ |
| description | Let $\mathbb{N}$ be the set of all nonnegative integers. For any positive integer $k$ and any subset $A$ of nonnegative integers, let $r_{1,k}(A,n)$ be the number of solutions $(a_1,a_2)$ to the equation $n=a_1+ka_2$. In 2016, Qu proved that
\[ \liminf _{n\,\rightarrow \,\infty }r_{1,k}(A,n)=\infty \]
providing that $r_{1,k}(A,n)=r_{1,k}(\mathbb{N}\setminus A,n)$ for all sufficiently large integers, which answered affirmatively a 2012 problem of Yang and Chen. In a very recent article, another Chen (the first named author) slightly improved Qu’s result and obtained that
\[ \liminf _{n\,\rightarrow \,\infty }\frac{r_{1,k}(A,n)}{\log n}>0. \]
In this note, we further improve the lower bound on $r_{1,k}(A,n)$ by showing that
\[ \liminf _{n\,\rightarrow \,\infty }\frac{r_{1,k}(A,n)}{n}>0. \]
Our bound reflects the correct order of magnitude of the representation function $r_{1,k}(A,n)$ under the above restrictions due to the trivial fact that $r_{1,k}(A,n)\le n/k.$ |
| format | Article |
| id | doaj-art-9ded25a3590d43709522fe8dff3aea68 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-05-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-9ded25a3590d43709522fe8dff3aea682025-02-07T11:21:12ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-05-01362G554755210.5802/crmath.57310.5802/crmath.573Correct order on some certain weighted representation functionsChen, Shi-Qiang0Ding, Yuchen1Lü, Xiaodong2Zhang, Yuhan3School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, People’s Republic of ChinaSchool of Mathematical Sciences, Yangzhou University, Yangzhou 225002, People’s Republic of ChinaSchool of Mathematical Sciences, Yangzhou University, Yangzhou 225002, People’s Republic of ChinaSchool of Mathematical Sciences, Yangzhou University, Yangzhou 225002, People’s Republic of ChinaLet $\mathbb{N}$ be the set of all nonnegative integers. For any positive integer $k$ and any subset $A$ of nonnegative integers, let $r_{1,k}(A,n)$ be the number of solutions $(a_1,a_2)$ to the equation $n=a_1+ka_2$. In 2016, Qu proved that \[ \liminf _{n\,\rightarrow \,\infty }r_{1,k}(A,n)=\infty \] providing that $r_{1,k}(A,n)=r_{1,k}(\mathbb{N}\setminus A,n)$ for all sufficiently large integers, which answered affirmatively a 2012 problem of Yang and Chen. In a very recent article, another Chen (the first named author) slightly improved Qu’s result and obtained that \[ \liminf _{n\,\rightarrow \,\infty }\frac{r_{1,k}(A,n)}{\log n}>0. \] In this note, we further improve the lower bound on $r_{1,k}(A,n)$ by showing that \[ \liminf _{n\,\rightarrow \,\infty }\frac{r_{1,k}(A,n)}{n}>0. \] Our bound reflects the correct order of magnitude of the representation function $r_{1,k}(A,n)$ under the above restrictions due to the trivial fact that $r_{1,k}(A,n)\le n/k.$https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.573/representation functionsorder of functionspartitions of integers |
| spellingShingle | Chen, Shi-Qiang Ding, Yuchen Lü, Xiaodong Zhang, Yuhan Correct order on some certain weighted representation functions Comptes Rendus. Mathématique representation functions order of functions partitions of integers |
| title | Correct order on some certain weighted representation functions |
| title_full | Correct order on some certain weighted representation functions |
| title_fullStr | Correct order on some certain weighted representation functions |
| title_full_unstemmed | Correct order on some certain weighted representation functions |
| title_short | Correct order on some certain weighted representation functions |
| title_sort | correct order on some certain weighted representation functions |
| topic | representation functions order of functions partitions of integers |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.573/ |
| work_keys_str_mv | AT chenshiqiang correctorderonsomecertainweightedrepresentationfunctions AT dingyuchen correctorderonsomecertainweightedrepresentationfunctions AT luxiaodong correctorderonsomecertainweightedrepresentationfunctions AT zhangyuhan correctorderonsomecertainweightedrepresentationfunctions |