REPRESENTATION FUNCTIONS ON ABELIAN GROUPS

Research paper by WU-XIA MA, YONG-GAO CHEN

Indexed on: 11 Jan '19Published on: 01 Feb '19Published in: Bulletin of the Australian Mathematical Society

Abstract

Let $G$ be a finite abelian group, $A$ a nonempty subset of $G$ and $h\geq 2$ an integer. For $g\in G$ , let $R_{A,h}(g)$ denote the number of solutions of the equation $x_{1}+\cdots +x_{h}=g$ with $x_{i}\in A$ for $1\leq i\leq h$ . Kiss et al. [‘Groups, partitions and representation functions’, Publ. Math. Debrecen85(3) (2014), 425–433] proved that (a) if $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for all $g\in G$ , then $; G; =2; A;$ , and (b) if $h$ is even and $; G; =2; A;$ , then $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for all $g\in G$ . We prove that $R_{G\setminus A,h}(g)-(-1)^{h}R_{A,h}(g)$ does not depend on $g$ . In particular, if $h$ is even and $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for some $g\in G$ , then $; G; =2; A;$ . If $h>1$ is odd and $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for all $g\in G$ , then $R_{A,h}(g)=\frac{1}{2}; A; ^{h-1}$ for all $g\in G$ . If $h>1$ is odd and $; G;$ is even, then there exists a subset $A$ of $G$ with $; A; =\frac{1}{2}; G;$ such that $R_{A,h}(g)\not =R_{G\setminus A,h}(g)$ for all $g\in G$ .