Multiplicative dependence of rational values modulo approximate finitely generated groups

In this paper, we establish some finiteness results about the multiplicative dependence of rational values modulo sets which are `close' (with respect to the Weil height) to division groups of finitely generated multiplicative groups of a number field $K$. For example, we show that under some conditions on polynomials $f_1, \ldots, f_n\in K[X]$, there are only finitely many elements $\alpha \in K$ such that $f_1(\alpha),\ldots,f_n(\alpha)$ are multiplicatively dependent modulo such sets. As one of our main tools, we prove a new version of the well-known Schinzel-Tijdeman theorem regarding $S$-integer solutions of hyper-elliptic equations for polynomials with at least two distinct roots.