Quantitative growth of linear recurrences

TL;DR

The paper establishes an explicit, effective bound on the number of indices $n$ for which a non-degenerate linear recurrence sequence with Binet form $u_n=\sum_{i=1}^{m} P_i(n)\alpha_i^n$ satisfies $|u_n|< (\max_i|\alpha_i|)^{n(1-\varepsilon)}$, for $\varepsilon$ in $(0,1/12)$. It blends geometry-of-numbers methods in number fields (heights, $S$-integers, successive minima) with Roth-type diophantine approximation to show that large-$n$ solutions lie in a finite union of proper $K$-subspaces and then quantifies this structure via effective bounds. The work extends prior qualitative results (Schmidt–Evertse, Schmidt, Schlickewei, Van der Poorten) by providing an explicit bound on the number of solutions, with implications for positivity/ultimate-positivity problems in linear recurrences. The results have theoretical significance for effective Diophantine analysis of recurrences and practical impact on decidability questions in related computational problems.

Abstract

Let $\{u_n\}_n$ be a non-degenerate linear recurrence sequence of integers with Binet's formula given by $u_n= \sum_{i=1}^{m} P_i(n)α_i^n.$ Assume $\max_i \vert α_i \vert >1$. In 1977, Loxton and Van der Poorten conjectured that for any $ε>0$ there is a effectively computable constant $C(ε),$ such that if $ \vert u_n \vert < (\max_i\{ \vert α_i \vert \})^{n(1-ε)}$, then $n<C(ε)$. 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 $\vert u_n \vert < (\max_i\{ \vert α_i \vert \})^{n(1-ε)}$, thus extending several earlier results by Schmidt, Schlickewei and Van der Poorten.

Theorems & Definitions (35)

• Theorem 1
• Lemma 2
• Remark 3
• Theorem 4
• Lemma 5
• Lemma 6
• proof
• Lemma 7
• proof
• Lemma 8