Additive and subtractive bases of $ \mathbb{Z}_m$ in average
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| Format: | Preprint |
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2024
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| _version_ | 1866916318109761536 |
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| author | Liang, Guangping Zhang, Yu Zuo, Haode |
| author_facet | Liang, Guangping Zhang, Yu Zuo, Haode |
| contents | Given a positive integer $m$, let $\mathbb{Z}_m$ be the set of residue classes mod $m$. For $A\subseteq \mathbb{Z}_m$ and $n\in \mathbb{Z}_m$, let $σ_A(n)$ be the number of solutions to the equation $n=x+y$ with $x,y\in A$. Let $\mathcal{H}_m$ be the set of subsets $A\subseteq \mathbb{Z}_m$ such that $σ_A(n)\geq1$ for all $n\in \mathbb{Z}_m$. Let
$$
\ell_m=\min\limits_{A\in \mathcal{H}_m}\left\lbrace m^{-1}\sum_{n\in \mathbb{Z}_m}σ_A(n)\right\rbrace.
$$ Following a prior result of Ding and Zhao on Ruzsa's number, we know that $$ \limsup_{m\rightarrow\infty}\ell_m\le 192. $$ Ding and Zhao then asked possible improvements on this value. In this paper, we prove $$ \limsup\limits_{m\rightarrow\infty}\ell_m\leq 144. $$ Moreover, parallel results on subtractive bases of $ \mathbb{Z}_m$ were also investigated here. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_02344 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Additive and subtractive bases of $ \mathbb{Z}_m$ in average Liang, Guangping Zhang, Yu Zuo, Haode Number Theory Given a positive integer $m$, let $\mathbb{Z}_m$ be the set of residue classes mod $m$. For $A\subseteq \mathbb{Z}_m$ and $n\in \mathbb{Z}_m$, let $σ_A(n)$ be the number of solutions to the equation $n=x+y$ with $x,y\in A$. Let $\mathcal{H}_m$ be the set of subsets $A\subseteq \mathbb{Z}_m$ such that $σ_A(n)\geq1$ for all $n\in \mathbb{Z}_m$. Let $$ \ell_m=\min\limits_{A\in \mathcal{H}_m}\left\lbrace m^{-1}\sum_{n\in \mathbb{Z}_m}σ_A(n)\right\rbrace. $$ Following a prior result of Ding and Zhao on Ruzsa's number, we know that $$ \limsup_{m\rightarrow\infty}\ell_m\le 192. $$ Ding and Zhao then asked possible improvements on this value. In this paper, we prove $$ \limsup\limits_{m\rightarrow\infty}\ell_m\leq 144. $$ Moreover, parallel results on subtractive bases of $ \mathbb{Z}_m$ were also investigated here. |
| title | Additive and subtractive bases of $ \mathbb{Z}_m$ in average |
| topic | Number Theory |
| url | https://arxiv.org/abs/2407.02344 |