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

Abstract

AbstractLet $$ G_n $$ G n and $$ H_m $$ H m be two non-degenerate linear recurrence sequences defined over a function field F in one variable over $$ \mathbb {C}$$ C , and let $$ \mu $$ μ be a valuation on F. We prove that under suitable conditions there are effectively computable constants $$ c_1 $$ c 1 and $$ C' $$ C ′ such that the bound $$\begin{aligned} \mu (G_n - H_m) \le \mu (G_n) + C' \end{aligned}$$ μ ( G n - H m ) ≤ μ ( G n ) + C ′ holds for $$ \max \left( n,m \right) > c_1 $$ max n , m > c 1 .