Common values of a class of linear recurrence

TL;DR

This paper investigates the common values of two linear recursive sequences with specific dominant roots, establishing effective bounds on their differences under certain algebraic conditions.

Contribution

It provides new effective bounds on the differences of two linear recurrence sequences with specified dominant roots and algebraic independence conditions.

Findings

Established bounds for the difference |a_n - b_m| under given conditions.

Proved that the sequences rarely take the same or close values.

Derived effective constants for the inequalities involved.

Abstract

Let $(a_n), (b_n)$ be linear recursive sequences of integers with characteristic polynomials $A(X),B(X)\in \mathbb{Z}[X]$ respectively. Assume that $A(X)$ has a dominating and simple real root $\alpha$, while $B(X)$ has a pair of conjugate complex dominating and simple roots $\beta,\bar{\beta}$. Assume further that $\alpha/ \beta$ and $\bar{\beta}/\beta$ are not roots of unity and $\delta = \log |\alpha|/ \log |\beta| \in \mathbb{Q}$. Then there are effectively computable constants $c_0,c_1>0$ such that the inequality $$ |a_n - b_m| > |a_n|^{1-(c_0 \log^2 n)/n} $$ holds for all $n,m \in \mathbb{Z}^2_{\ge 0}$ with $\max\{n,m\}>c_1$.