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Transactions of the American Mathematical Society

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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
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Abstract: For every complex number $ x$, let $ \Vert x\Vert _{\mathbb{Z}}:=\min \{\vert x-m\vert:\ 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
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
Email: dangkhoa.nguyen@ucalgary.ca

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