Common values of a class of linear recurrence

Abstract

Let (an), (bn) be linear recursive sequences of integers with characteristic polynomials A(X), B(X) ∈ Z[X] respectively. Assume that A(X) has a dominating and simple real root α, while B(X) has a pair of conjugate complex dominating and simple roots β, β. Assume further that α/β and β/β are not roots of unity and δ = log |α|/ log |β| ∈ Q. Then there are effectively computable constants c 0 , c 1 > 0 such that the inequalityholds for all n, m ∈ Z 2 ≥0 with max{n, m} > c 1 .