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}$}:=\min{|x-m|:
m\in$\mathbb{Z}$}$\mathrm{.}Let$K$beanumberfield,let$k\in$\mathbb{N}$$,andlet$\alpha_1,\ldots,\alpha_k$ be non-zero algebraic numbers. In this paper, we completely solve the problem of the existence of $\theta \in (0,1)$ such that there are infinitely many tuples $(n,q_1,\dots ,q_k)$ satisfying $\Vert q_1\alpha_1^n+\ldots+q_k\alpha_k^n\Vert_{$\mathbb{Z}$}<\theta^n$where$n\in$\mathbb{N}$$and$q_1,\ldots,q_k\in K^*$ having small logarithmic height compared to $n$. In the special case when $q_1,\dots ,q_k$ have the form $q_i=q{c}_i$ for fixed $c_1,\dots ,c_k$, our work yields results on algebraic approximations of $c_1{\alpha }_1^n+\dots +c_k{\alpha }_k^n$ of the form ${\displaystyle \frac{m}{q}}$ with $m\in \mathbb{Z}$ and $q\in K^{\ast }$ (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.