  For every complex number $x$, let $\Vert x\Vert_{\mathbb{Z}}:=\min\{|x-m|:\
m\in\mathbb{Z}\}$. 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 $\Vert
q_1\alpha_1^n+\ldots+q_k\alpha_k^n\Vert_{\mathbb{Z}}<\theta^n$ where
$n\in\mathbb{N}$ and $q_1,\ldots,q_k\in K^*$ having small logarithmic height
compared to $n$. In the special case when $q_1,\ldots,q_k$ have the form
$q_i=qc_i$ for fixed $c_1,\ldots,c_k$, our work yields results on algebraic
approximations of $c_1\alpha_1^n+\ldots+c_k\alpha_k^n$ of the form
$\displaystyle \frac{m}{q}$ with $m\in \mathbb{Z}$ and $q\in K^*$ (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 $q_1$ 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 play an
important role in the proof of our results.
