Algebraic approximations to linear combinations of powers: An extension of results by mahler and Corvaja–Zannier

For every complex number x, let‖x‖ _Z := min{|x − m|: m ∈ Z}. Let K be a number field, let k ∈ N, andletα ₁ ,…,α _k be non-zero algebraic numbers. In this paper, we completely solve the problem of the existence of θ ∈ (0, 1) such that there are infinitely many tuples (n, q ₁ ,…,q _k ) satisfying ‖q ₁ α ⁿ ₁ + ··· + q _k α ⁿ _k ‖ _Z < θ ⁿ where n ∈ N and q ₁ ,…,q _k ∈ K ^∗ have small logarithmic height compared to n. In the special case when q ₁ ,…,q _k have the form q _i = qc _i for fixed c ₁ ,…,c _k , our work yields results on algebraic approximations of c ₁ α ⁿ ₁ + ···+ c _k α ⁿ _k of the form ^m _q with m ∈ Z and q ∈ K ^∗ (where q has small logarithmic height compared to n). Various results on linear recurrence sequences also follow as an immediate consequence. The case where k =1andq ₁ is essentially a rational integer was obtained by Corvaja and Zannier and settled a long-standing question of Mahler. The use of the Subspace Theorem based on work of Corvaja–Zannier, together with several modifications, plays an important role in the proof of our results.