On the Existence of Perfect Splitter Sets

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.