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$\mathbb {Z}_m$
WITH IDENTICAL REPRESENTATION FUNCTION
For a given set
$S\subseteq \mathbb {Z}_m$
and
$\overline {n}\in \mathbb {Z}_m$
, let
$R_S(\overline {n})$
denote the number of solutions of the equation
$\overline {n}=\overline {s}+\overline {s'}$
with ordered pairs
$(\overline {s},\overline {s'})\in S^2$
. We determine the structure of
$A,B\subseteq \mathbb {Z}_m$
with
$|(A\cup B)\setminus (A\cap B)|=m-2$
such that
$R_{A}(\overline {n})=R_{B}(\overline {n})$
for all
$\overline {n}\in \mathbb {Z}_m$
, where m is an even integer.
