Asymptotic formulas for sums of elements from a multiplicative group

Let $K$ be a number field, $k\ge 2$ an integer, $(K^{\ast }{)}^k$ the $k$-fold direct product of $K^{\ast }$ with coordinatewise multiplication, and $\mathrm{\Gamma }$ a finitely generated subgroup of rank $r$ of $(K^{\ast }{)}^k$. Further, let $H(\alpha )$ denote the absolute exponential height of an algebraic number $\alpha$. Fix non-zero elements $a_1,\dots ,a_k\in K$. We give asymptotic formulas for the number of $\mathbf{x}=(x_1,\dots ,x_k)\in \mathrm{\Gamma }$ with $H(a_1{x}_1+\cdots +a_k{x}_k)\le X$ as $X\to \mathrm{\infty }$ such that no non-empty subsum of $a_1{x}_1+\cdots +a_k{x}_k$ vanishes. By the same method of proof, we obtain an asymptotic formula as $X\to \mathrm{\infty }$ for the number of non-negative integers $n$ with $H(u_n)\le X$, where $\{u_n\}$ is a linear recurrence sequence.