ON ADDITIVE REPRESENTATION FUNCTIONS
ON ADDITIVE REPRESENTATION FUNCTIONS
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DOI：
10.1017/s0004972717000302
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发表时间：
2017-05
影响因子：
0.7
通讯作者：
Ya-Li Li-Ya-Li-Li-2110479499;Yong-Gao Chen
中科院分区：
文献类型：
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作者：
Ya-Li Li-Ya-Li-Li-2110479499;Yong-Gao Chen
对于任意有限交换群$G$，其中$|G|=m$，$A子集G$和$g\在G$中，设$R_{A}(G)$是方程$g=a+b$，$a，b在A$中的解的个数.最近，S等人证明了：如果$m36$和$R_{A}(N)1$对ő{Z}_{m}$中的所有$n都有$n，则在Mathbb{Z}_{m}$中存在$n\in{Z}_{m}$使得$R_{A}(N)\geq 6$。本文证明了：(A)对于任意具有$|G|=m$和$A子集G$的有限交换群$G$，如果$R_{A}(G)=0$的$g的个数不超过$\FRAC{7}{32}m-\FRAC{1}{2}\Sqrt{10m}-1$，则G中存在$G\使得$R_{A}(G)\geq 6$；(B)如果对G$中的所有$g都有$1\leq R_{A}(G)\leq 6$，则具有$R_{A}(G)=6$的G$中的$g的个数大于$\frac{7}{32}m-\frac{1}{2}\sqrt{10m}-1$。
For any finite abelian group $G$ with $|G|=m$ , $A\subseteq G$ and $g\in G$ , let $R_{A}(g)$ be the number of solutions of the equation $g=a+b$ , $a,b\in A$ . Recently, Sándor and Yang [‘A lower bound of Ruzsa’s number related to the Erdős–Turán conjecture’, Preprint, 2016, arXiv:1612.08722v1] proved that, if $m\geq 36$ and $R_{A}(n)\geq 1$ for all $n\in \mathbb{Z}_{m}$ , then there exists $n\in \mathbb{Z}_{m}$ such that $R_{A}(n)\geq 6$ . In this paper, for any finite abelian group $G$ with $|G|=m$ and $A\subseteq G$ , we prove that (a) if the number of $g\in G$ with $R_{A}(g)=0$ does not exceed $\frac{7}{32}m-\frac{1}{2}\sqrt{10m}-1$ , then there exists $g\in G$ such that $R_{A}(g)\geq 6$ ; (b) if $1\leq R_{A}(g)\leq 6$ for all $g\in G$ , then the number of $g\in G$ with $R_{A}(g)=6$ is more than $\frac{7}{32}m-\frac{1}{2}\sqrt{10m}-1$ .