# REPRESENTATION FUNCTIONS ON ABELIAN GROUPS

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

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

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#### 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
$g\in G$
.