Quantitative growth of linear recurrences

Let $\{u_n{\}}_n$ be a non-degenerate linear recurrence sequence of integers with Binet’s formula given by $u_n={\sum }_{i=1}^m{P}_i(n){\alpha }_i^n\mathrm{.}$ Assume ${\max }_i\mathrm{\mid }{\alpha }_i\mathrm{\mid }>1$. In 1977, Loxton and Van der Poorten conjectured that for any $\epsilon >0$ there is a effectively computable constant $C(\epsilon ),$ such that if $\mathrm{\mid }{u}_n\mathrm{\mid }<({\max }_i\{\mathrm{\mid }{\alpha }_i\mathrm{\mid }\}{)}^{n(1-\epsilon )}$, 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 $\mathrm{\mid }{u}_n\mathrm{\mid }<({\max }_i\{\mathrm{\mid }{\alpha }_i\mathrm{\mid }\}{)}^{n(1-\epsilon )}$, thus extending several earlier results by Schmidt, Schlickewei and Van der Poorten.