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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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2024-05-01
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| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.573/ |
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| Summary: | 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 $r_{1,k}(A,n)\le n/k.$ |
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| ISSN: | 1778-3569 |