  Given integers $k_1, k_2$ with $0\le k_1<k_2$, the determinations of all
positive integers $q$ for which there exists a perfect Splitter $B[-k_1,
k_2](q)$ set is a wide open question in general. In this paper, we obtain new
necessary and sufficient conditions for an odd prime $p$ such that there exists
a nonsingular perfect $B[-1,3](p)$ set. We also give some necessary conditions
for the existence of purely singular perfect splitter sets. In particular, we
determine all perfect $B[-k_1, k_2](2^n)$ sets for any positive integers
$k_1,k_2$ with $k_1+k_2\ge4$. We also prove that there are infinitely many
prime $p$ such that there exists a perfect $B[-1,3](p)$ set.
