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

Authors:
Avinash Kulkarni, Niki Myrto Mavraki and Khoa D. Nguyen

Journal:
Trans. Amer. Math. Soc. **371** (2019), 3787-3804

MSC (2010):
Primary 11J68, 11J87; Secondary 11B37, 11R06

DOI:
https://doi.org/10.1090/tran/7316

Published electronically:
November 16, 2018

MathSciNet review:
3917208

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Abstract | References | Similar Articles | Additional Information

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.

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Additional Information

**Avinash Kulkarni**

Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada

Email:
akulkarn@sfu.ca

**Niki Myrto Mavraki**

Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada

Address at time of publication:
Department of Mathematics, Northwestern University, Evanston, Illinois 60208

MR Author ID:
1050391

Email:
myrtomav@northwestern.edu

**Khoa D. Nguyen**

Affiliation:
Department of Mathematics, University of British Columbia — and — Pacific Institute for The Mathematical Sciences, Vancouver, British Columbia V6T 1Z2, Canada

Address at time of publication:
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta T2N 1N4, Canada

MR Author ID:
886774

Email:
dangkhoa.nguyen@ucalgary.ca

Keywords:
Algebraic approximations,
linear combinations of powers,
linear recurrence sequences,
subspace theorem

Received by editor(s):
February 25, 2017

Received by editor(s) in revised form:
May 24, 2017, and June 20, 2017

Published electronically:
November 16, 2018

Additional Notes:
The first author was partially supported by NSERC

The third author was partially supported by a UBC-PIMS fellowship

Article copyright:
© Copyright 2018
American Mathematical Society