On the size of a linear combination of two linear recurrence sequences over function fields

Abstract: Let $ G_n $ and $ H_m $ be two non-degenerate linear recurrence sequences defined over a function field $ F $ in one variable over $ \mathbb{C} $, and let $ \mu $ be a valuation on $ F $. We prove that under suitable conditions there are effectively computable constants $ c_1 $ and $ C' $ such that the bound \begin{equation*} \mu(G_n - H_m) \leq \mu(G_n) + C' \end{equation*} holds for $ \max {n,m} > c_1 $.