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 ${\alpha }_1,\dots ,{\alpha }_k$ 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 $\Vert q_1{\alpha }_1^n+\dots +q_k{\alpha }_k^n{\Vert }_{\mathbb{Z}}<{\theta }^n$ where $n\in \mathbb{N}$ and $q_1,\dots ,q_k\in K^{\ast }$ 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 $\frac{m}{q}$ 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.

arXiv comment16 pages, comments are welcome