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循环群中的差基

Difference bases in cyclic groups

Taras Banakh, Volodymyr Gavrylkiv

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11 pages
发表信息：
J. Algebra and its Appl. 18:05 (2019) 1950081
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摘要：

阿贝尔群$G$的子集$B$称为$G$的差基，如果每个元素$g\in G$可以写为某些元素$a,b\in B$的差$g=a-b$。差分基$B\subset G$的最小基数$|B|$称为$G$的差分大小，并由$Δ[G]$表示。证明了对每个$n\in\mathbb N$，阶为$n$的循环群$C_n$具有差大小$\frac{1+\sqrt{4|n|-3}}2\le Δ[C_n]\le\frac32\sqrt{n}$ .如果是$n\ge 9$（和$n\ge 2\cdot 10^{15}$），则是$Δ[C_n]\le\frac{12}{\sqrt{73}}\sqrt{n}$（和$Δ[C_n]<\frac2{\sqrt{3}}\sqrt{n}$）。并计算了基数为$\le 100$的所有循环群的差的大小。

英文摘要：

A subset $B$ of an Abelian group $G$ is called a difference basis of $G$ if each element $g\in G$ can be written as the difference $g=a-b$ of some elements $a,b\in B$. The smallest cardinality $|B|$ of a difference basis $B\subset G$ is called the difference size of $G$ and is denoted by $Δ[G]$. We prove that for every $n\in\mathbb N$ the cyclic group $C_n$ of order $n$ has difference size $\frac{1+\sqrt{4|n|-3}}2\le Δ[C_n]\le\frac32\sqrt{n}$. If $n\ge 9$ (and $n\ge 2\cdot 10^{15}$), then $Δ[C_n]\le\frac{12}{\sqrt{73}}\sqrt{n}$ (and $Δ[C_n]<\frac2{\sqrt{3}}\sqrt{n}$). Also we calculate the difference sizes of all cyclic groups of cardinality $\le 100$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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