On unavoidable obstructions in Gaussian walks

Abstract

In this paper we investigate a problem about certain walks in the ring of Gaussian integers. Let $n,d$ be two natural numbers. Does there exist a sequence of Gaussian integers $z_j$ such that $|z_{j+1}-z_j|=1$ and a pair of indices $r$ and $s$, such that $z_{r}-z_{s}=n$ and for all indices $t$ and $u$, $z_{t}-z_{u}\neq d$? If there exists such a sequence we call $n$ to be $d$ avoidable. Let $A_n$ be the set of all $d\in \mathbb{N}$ such that $n$ is not $d$ avoidable. Recently, Ledoan and Zaharescu proved that $\{d \in \mathbb{N} : d|n\}\subset A_n$. We extend this result by giving a necessary and sufficient condition for $d\in A_n$ which answers a question posed by Ledoan and Zaharescu. We also find a precise formula for the cardinality of $A_n$ and answer three other questions raised in the same paper.