Multiplicative dependence modulo subsets

In this paper, we show that if the non-constant rational functions $f_1,\dots ,f_n\in K(x)$ over a number field $K$ cannot multiplicatively generate a power of a linear fractional function, then there are only finitely many elements $\alpha \in K$ such that $f_1(\alpha ),\dots ,f_n(\alpha )$ are multiplicatively dependent modulo some subset `close’ (with respect to the Weil height) to the division group of a finitely generated multiplicative subgroup of $K$. This improves some previous results.