Note on unique representation bases
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866917257328721920 |
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| author | Ding, Yuchen Wang, Jie |
| author_facet | Ding, Yuchen Wang, Jie |
| contents | Answering affirmatively a 2007 problem of Chen, the first author proved that there is a unique representation basis $A$ of $\mathbb{Z}$ and a constant $c>0$ such that $$ A(-x,x)\ge c\sqrt{x} $$ for infinitely many positive integers $x$, where $A(-x,x)=\big|A\cap[-x, x]\big|$. Let $c_{\mathscr{A}}$ be the least upper bound for such $c$. It was proved in the former article by the first author that $\sqrt{2}/2\le c_{\mathscr{A}}\le \sqrt{2}$. In this note, the prior result is improved to $c_{\mathscr{A}}\ge 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_07743 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Note on unique representation bases Ding, Yuchen Wang, Jie Number Theory Answering affirmatively a 2007 problem of Chen, the first author proved that there is a unique representation basis $A$ of $\mathbb{Z}$ and a constant $c>0$ such that $$ A(-x,x)\ge c\sqrt{x} $$ for infinitely many positive integers $x$, where $A(-x,x)=\big|A\cap[-x, x]\big|$. Let $c_{\mathscr{A}}$ be the least upper bound for such $c$. It was proved in the former article by the first author that $\sqrt{2}/2\le c_{\mathscr{A}}\le \sqrt{2}$. In this note, the prior result is improved to $c_{\mathscr{A}}\ge 1$. |
| title | Note on unique representation bases |
| topic | Number Theory |
| url | https://arxiv.org/abs/2602.07743 |