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 ${\max }_i|{\alpha }_i|>1$. In 1977, Loxton and Van der Poorten conjectured that for any $\epsilon >0$ there is a effectively computable constant $C(\epsilon ),$ such that if $|u_n|<({\max }_i\{|{\alpha }_i|\}{)}^{n(1-\epsilon )}$, 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 $|u_n|<({\max }_i\{|{\alpha }_i|\}{)}^{n(1-\epsilon )}$, thus extending several earlier results by Schmidt, Schlickewei and Van der Poorten.

32 pages

References in corpus (1)

• On the growth of linear recurrences in function fields