  Let $\Ga = (V, E)$ be a graph and $a, b$ nonnegative integers. An $(a,
b)$-regular set in $\Ga$ is a nonempty proper subset $D$ of $V$ such that every
vertex in $D$ has exactly $a$ neighbours in $D$ and every vertex in $V
\setminus D$ has exactly $b$ neighbours in $D$. A $(0,1)$-regular set is called
a perfect code, an efficient dominating set, or an independent perfect
dominating set. A subset $D$ of a group $G$ is called an $(a,b)$-regular set of
$G$ if it is an $(a, b)$-regular set in some Cayley graph of $G$, and an $(a,
b)$-regular set in a Cayley graph of $G$ is called a subgroup $(a, b)$-regular
set if it is also a subgroup of $G$. In this paper we study $(a, b)$-regular
sets in Cayley graphs with a focus on $(0, k)$-regular sets, where $k \ge 1$ is
an integer. Among other things we determine when a non-trivial proper normal
subgroup of a group is a $(0, k)$-regular set of the group. We also determine
all subgroup $(0, k)$-regular sets of dihedral groups and generalized
quaternion groups. We obtain necessary and sufficient conditions for a
hypercube or the Cartesian product of $n$ copies of the cycle of length $p$ to
admit $(0, k)$-regular sets, where $p$ is an odd prime. Our results generalize
several known results from perfect codes to $(0, k)$-regular sets.
