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 $n,m\in {\mathbb{Z}}_{\ge 0}^2$ with $\max \{n,m\}>c_1$.