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

TL;DR

This paper characterizes when algebraic linear combinations of powers can be approximated exponentially well by integers, extending classical results and employing the Subspace Theorem to solve longstanding problems in number theory.

Contribution

It provides a complete characterization of the existence of exponential approximations for algebraic linear combinations of powers, extending prior work by Mahler and Corvaja-Zannier.

Findings

Characterization of exponential approximation conditions

Extension of Mahler and Corvaja-Zannier results

Applications to linear recurrence sequences

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 $\Vert q_1\alpha_1^n+\ldots+q_k\alpha_k^n\Vert_{\mathbb{Z}}<\theta^n$ 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.