REPRESENTATION FUNCTIONS ON ABELIAN GROUPS

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$1\leq i\leq h$. Kisset 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$g\in G$.