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

For every complex number x, let \Vert x\Vert _{\mathbb{Z}}:=\min \{\vert x-m\vert:\ m\in \mathbb{Z}\}. Let K be a number field, let k\in \mathbb{N}, and let \alpha _1,\ldots ,\alpha _k be non-zero algebraic numbers. In this paper, we completely solve the problem of the existence of \theta \in (0,1) such that there are infinitely many tuples (n,q_1,\ldots ,q_k) satisfying \Vert q_1\alpha _1^n+\cdots +q_k\alpha _k^n\Vert _{\mathbb{Z}}