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Let $K$ be a field of characteristic $p > 0$ and let $G$ be a subgroup of$K^\ast \times K^\ast$ with $\text{dim}_\mathbb{Q}(G \otimes_\mathbb{Z}\mathbb{Q}) = r$ finite. Then Voloch proved that the equation $ax + by = 1\text{ in } (x, y) \in G$ for given $a, b \in K^\ast$ has at most $p^r(p^r + p- 2)/(p - 1)$ solutions $(x, y) \in G$, unless $(a, b)^n \in G$ for some $n\geq 1$.  Voloch also conjectured that this upper bound can be replaced by onedepending only on $r$. Our main theorem answers this conjecture positively. Weprove that there are at most $31 \cdot 19^{r + 1}$ solutions $(x, y)$ unless$(a, b)^n \in G$ for some $n \geq 1$ with $(n, p) = 1$.  During the proof of our main theorem we generalize the work of Beukers andSchlickewei to positive characteristic, which heavily relies on diophantineapproximation methods. This is a surprising feat on its own, since usuallythese methods can not be transferred to positive characteristic.

On the equation $x + y = 1$ in finitely generated groups in positive characteristic