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

Abstract:

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 $\Vert q_1\alpha _1^n+\cdots +q_k\alpha _k^n\Vert _{\mathbb {Z}}<\theta ^n$ where $n\in \mathbb {N}$ and $q_1,\ldots ,q_k\in K^*$ have 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 $c_1\alpha _1^n+\cdots +c_k\alpha _k^n$ of the form $\frac {m}{q}$ 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 where $k=1$ 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, plays an important role in the proof of our results.