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ON ADDITIVE REPRESENTATION FUNCTIONS

  • YA-LI LI (a1) and YONG-GAO CHEN (a2)

Abstract

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 $\frac{7}{32}m-\frac{1}{2}\sqrt{10m}-1$ .

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This work was supported by the National Natural Science Foundation of China, Grant No. 11371195, and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

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References

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