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 rational functions f₁, …, fₙ∈ K(X) , there are only finitely many elements α ∈ K such that f₁(α),…,fₙ(α) are multiplicatively dependent modulo such sets.