Weak uniform distribution of \(u_{n+1}=au_ n+b\) in Dedekind domains (Q1106873)

Let \(R\) be a Dedekind domain and \(I\) a non-trivial ideal of it. The authors consider sequences \((u_ n)\) in \(R\), satisfying the recurrence \(u_{n+1}=au_ n+b\) and give necessary and sufficient conditions (in terms of \(a, b, u_ 0)\) for such a sequence to be weakly uniformly distributed (mod \(I\)). As a corollary they obtain that such a sequence is weakly uniformly distributed (mod \(I\)) if and only if the congruence \(u_ x\equiv a\pmod I\) is solvable for every \(a\) prime to \(I\).