On $\mathfrak{P}$-adic continued fractions with extraneous denominators: some explicit finiteness results

Let $K$ be a number field. We show that, up to allowing a finite set of denominators in the partial quotients, it is possible to define algorithms for $\mathfrak{P}$-adic continued fractions satisfying the finiteness property on $K$ for every prime ideal $\mathfrak{P}$ of sufficiently large norm. This provides, in particular, a new algorithmic approach to the construction of division chains in number fields.