Common values of a class of linear recurrence

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 $\max\{n,m\}>c_1$.