On the Existence of Perfect Splitter Sets

TL;DR

This paper investigates the existence of perfect splitter sets, providing new conditions for certain cases, characterizing sets for powers of two, and proving infinitely many primes admit such sets.

Contribution

It establishes new necessary and sufficient conditions for perfect splitter sets, especially for odd primes and powers of two, advancing understanding of their existence.

Findings

Characterization of perfect B[-1,3](p) sets for odd primes p

Complete determination of perfect B[-k_1,k_2](2^n) sets for k_1+k_2≥4

Proof of infinitely many primes p with perfect B[-1,3](p) sets

Abstract

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.