On the Existence of Perfect Splitter Sets

Pingzhi Yuan; Kevin Zhao

paper

On the Existence of Perfect Splitter Sets

arXiv:1903.00118

Abstract

Given integers $k_{1}, k_{2}$ with $0 \leq k_{1} < k_{2}$ , 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 $k_{1} + k_{2} \geq 4$ . We also prove that there are infinitely many prime $p$ such that there exists a perfect $B [- 1, 3] (p)$ set.

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Some New Results on Splitter Sets