# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## Algebraic approximations to linear combinations of powers: An extension of results by Mahler and Corvaja–ZannierHTML articles powered by AMS MathViewer

by Avinash Kulkarni, Niki Myrto Mavraki and Khoa D. Nguyen
Trans. Amer. Math. Soc. 371 (2019), 3787-3804 Request permission

## 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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• 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
• 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