Let
$G$
be a commutative group,
$Y$
a real Banach space and
$f:G\rightarrow Y$
. We prove the Ulam–Hyers stability theorem for the cyclic functional equation
$$\begin{eqnarray}\displaystyle \frac{1}{|H|}\mathop{\sum }_{h\in H}f(x+h\cdot y)=f(x)+f(y) & & \displaystyle \nonumber\end{eqnarray}$$
for all
$x,y\in {\rm\Omega}$
, where
$H$
is a finite cyclic subgroup of
$\text{Aut}(G)$
and
${\rm\Omega}\subset G\times G$
satisfies a certain condition. As a consequence, we consider a measure zero stability problem of the functional equation
$$\begin{eqnarray}\displaystyle \frac{1}{N}\mathop{\sum }_{k=1}^{N}f(z+{\it\omega}^{k}{\it\zeta})=f(z)+f({\it\zeta}) & & \displaystyle \nonumber\end{eqnarray}$$
for all
$(z,{\it\zeta})\in {\rm\Omega}$
, where
$f:\mathbb{C}\rightarrow Y,\,{\it\omega}=e^{2{\it\pi}i/N}$
and
${\rm\Omega}\subset \mathbb{C}^{2}$
has four-dimensional Lebesgue measure
$0$
.