A divisibility obstruction for certain walks on Gaussian integers (Q2926283)

Let \(z_1,z_2,\dots\) be a finite or infinite sequence of Gaussian integers satisfying \(|z_{i+1}-z_i|=1\) and denote by \(A\) the set of all differences \(z_i-z_j\). The authors show (Theorem 1) that if \(n\) is a rational integer lying in \(A\), then each divisor \(d\) of \(n\) also lies in \(A\). This result is then generalized (Theorem 2) to linear spaces \(V\) over a field of characteristic zero as follows. Let \(u,v\in V\) be linearly independent, and let \(w_0,w_1,\dots,w_m\) be a sequence of elements of \(V\) with the following properties: there is a positive rational integer \(n\) such that \(w_m=w_0+nu\) and \(w_i-w_{i-1}\in\{\pm u,\pm v\}\) for \(i=1,2,\dots,m\). If \(d\) is a divisor of \(n\), then there exist \(i,j\) with \(w_i-w_j=du\).