Algebraic approximations to linear combinations of powers: an extension of results by Mahler and Corvaja-Zannier

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For every complex number x, let x:={|x-m|:m}. Let K be a number field, let k, and let α₁,,αₖ 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,q₁,,qₖ) satisfying q₁α₁ⁿ++qₖαₖⁿ<θⁿ where n and q₁,,qₖ∈ K* having small logarithmic height compared to n. In the special case when q₁,,qₖ have the form qᵢ=qcᵢ for fixed c₁,,cₖ, our work yields results on algebraic approximations of c₁α₁ⁿ++cₖαₖⁿ 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 k=1 and q₁ 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 play an important role in the proof of our results.