Multiplicative dependence in the sumset of multiplicative groups

Abstract

Let $\hat{I}“$ and $\hat{I}”$ be finitely generated multiplicative groups of algebraic numbers such that $\hat{I}“\cap \hat{I}”$ is a finite group. We show that, up to finitely many exceptions, non-zero sums $x_1+y_1$ and $x_2+y_2$, with $x_1,x_2\in \hat{I}“$ and $y_1,y_2\in \hat{I}”$, are multiplicatively dependent only if $x_1/x_2=y_1/y_2$ is a root of unity. For $m\ge 3$, we discuss possible shapes of $m$ multiplicatively dependent sums $x_1+y_1,\dots ,x_m+y_m$ with $x_1,\dots ,x_m\in \hat{I}“$ and $y_1,\dots ,y_m\in \hat{I}”$. For $m=3$ we classify such sums, up to finitely many exceptions, assuming the $abc$-conjecture.

Preliminary version, comments are welcome