On Unavoidable Obstructions in Gaussian Walks

We investigate a problem about certain walks in the ring of Gaussian integers. Let n and d be two natural numbers. Does there exist a sequence of Gaussian integers zj , such that |zj+1 zj | = 1 and a pair of indices r and s, such that zr zs = n and for all indices t and u, zt zu 6= d? If there exists such a sequence, we say that n is d avoidable. Let An be the set of all d 2 N such that n is not d avoidable. Recently, Ledoan and Zaharescu proved that {d 2 N : d|n} ⇢ An. We extend this result by giving a necessary and su cient condition for d 2 An, which answers a question posed by Ledoan and Zaharescu. We also find a precise formula for the cardinality of An and answer three other questions raised in the same paper.