On unavoidable obstructions in Gaussian walks

TL;DR

This paper studies the existence of specific Gaussian integer walks with constraints on differences, characterizes avoidable differences, and provides formulas for their sets' sizes, extending previous results in number theory.

Contribution

It offers a necessary and sufficient condition for when a difference is in the set of non-avoidable differences, advancing understanding of Gaussian walks with difference constraints.

Findings

Characterization of avoidable differences for Gaussian walks

A formula for the size of the set of non-avoidable differences

Extension of previous results by Ledoan and Zaharescu

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.