Quantitative growth of linear recurrences

Abstract

Let $\{u_n\}_n$ be a non-degenerate linear recurrence sequence of integers with Binet's formula given by $u_n= \sum_{i=1}^{m} P_i(n)\alpha_i^n.$ Assume $\max_i \vert \alpha_i \vert >1$. In 1977, Loxton and Van der Poorten conjectured that for any $\epsilon >0$ there is a effectively computable constant $C(\epsilon),$ such that if $ \vert u_n \vert < (\max_i\{ \vert \alpha_i \vert \})^{n(1-\epsilon)}$, then $n<C(\epsilon)$. Using results of Schmidt and Evertse, a complete non-effective (qualitative) proof of this conjecture was given by Fuchs and Heintze (2021) and, independently, by Karimov and al.~(2023). In this paper, we give an effective upper bound for the number of solutions of the inequality $\vert u_n \vert < (\max_i\{ \vert \alpha_i \vert \})^{n(1-\epsilon)}$, thus extending several earlier results by Schmidt, Schlickewei and Van der Poorten.