The Skolem Problem asks to determine whether a given integer linear recurrence sequence has a zero term. This problem, whose decidability has been open for 90 years, arises across a wide range of topics in computer science and dynamical system.  In 1977, a generalization of this problem was made by Loxton and Van der Poorten who conjectured that for any \(\epsilon >0\) and \(\{u_n\}\) a linear recurrence sequence with dominant (s) roots \(>1\) in absolute value, 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 talk, we prove a quantitative version of that result by giving an explicit upper bound for the number of solutions. Moreover, we give a function field analogue on growth of multi-recurrence, answering a question posed by Fuchs and Heintze when proving a bound on the growth of linear recurrences in function fields and generalizing a result of Fuchs and Pethö.

This talk is based on my paper accepted for publication in "Journ. Australia Math. Soc."  and an ongoing joint work with Clemens Fuchs.