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ON ADDITIVE REPRESENTATION FUNCTIONS

Part of: Sequences and sets

Published online by Cambridge University Press:  02 May 2017

YA-LI LI  and
YONG-GAO CHEN Open the ORCID record for YONG-GAO CHEN [Opens in a new window]
YA-LI LI
Affiliation:
School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, PR China email [email protected]
YONG-GAO CHEN*
Affiliation:
School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, PR China email [email protected]
*
✉[email protected]
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Abstract

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

Keywords

Erdős–Turán conjectureRuzsa’s numberrepresentation functionadditive basisabelian group

MSC classification

Primary: 11B34: Representation functions
Secondary: 11B13: Additive bases, including sumsets
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 96 , Issue 3 , December 2017 , pp. 380 - 388
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

Footnotes

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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